Finite Group Databasebeta

← Prev (#72283810) Next (#72283812) →

#72283811

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237593
commutator#2309C22×C4
Frattini, cosocle#20586C23
prime radical#72283717C2
LCSG ▷ #2309 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #6432959 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

3286 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 3286 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,13)(4,9,20,31)(5,21)(6,11,25,35)(7,12)(8,15)(10,22)(14,36,41,58)(16,44)(17,38)(18,29,23,33)(19,30)(24,34,52,51)(26,39,32,47)(27,42)(28,43)(37,46)(40,61,63,60)(45,54)(48,56)(49,55)(50,57)(53,62,59,64);(3,14)(5,22)(8,26)(13,36)(15,39)(16,40)(17,41)(18,23)(27,53)(28,32)(34,51)(37,61)(38,58)(42,64)(43,47)(44,60)(45,62)(46,63)(54,59)(55,56);(1,3,6,16)(2,8,11,27,12,28,35,54)(4,15,19,42,20,43,50,45)(5,14,24,40,22,41,52,63)(7,17,25,46)(9,13,30,37)(10,32,34,59)(18,47,48,64)(21,26,51,53)(23,39,49,62)(29,36,55,60,33,58,56,61)(31,38,57,44)

Polycyclic presentation:

2^11|a^2=gi,c^2=f,d^2=hij,e^2=hik,f^2=hj,b^a=bg,c^a=cd,c^b=ce,d^a=dh,d^c=dg,e^a=eij,e^b=ehik,e^c=eghij,e^d=ejk,f^a=fghi,f^b=fg,f^d=fi,f^e=fij,g^a=gj,g^c=gi,g^d=gj,g^e=gjk,g^f=gj,h^a=hjk,h^b=hj,h^c=hk,h^d=hk,h^e=hk,i^a=ij,i^b=ij,i^c=ij,i^d=ik,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285855, #72286183, #72286190
maximal normal subgroup of#72285855, #72286183, #72286190

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7*F.9);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.8*F.9*F.11);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);