Finite Group Databasebeta

← Prev (#72237935) Next (#72237937) →

#72237936

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72235864
commutator#2308C22×C4
Frattini, cosocleC2×#2308C23
prime radical#72237871C2
LCSG ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C26 ◁ C2×#2308 ◁ G
LECSG ▷ C2×#2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1

3856 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 3856 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(2,7,9,21)(3,11,13,30)(4,14)(5,17)(8,23,25,32)(10,27,18,22)(12,31,15,29)(20,28,24,26);(1,2)(3,8)(4,7)(5,18)(6,9)(10,17)(11,26)(12,20)(13,25)(14,21)(15,24)(16,22)(19,27)(23,31)(28,30)(29,32);(1,3)(2,8)(4,15)(5,11)(6,13)(7,20)(9,25)(10,28)(12,14)(16,31)(17,30)(18,26)(19,29)(21,24)(22,32)(23,27)

Polycyclic presentation:

2^11|a^2=e,d^2=i,f^2=ij,b^a=bd,c^a=cf,c^b=cg,d^a=dk,d^b=di,d^c=dh,e^b=eik,e^c=eijk,f^a=fk,f^b=fhi,f^c=fij,f^d=fjk,g^a=gij,g^d=gk,h^a=hj,h^d=hk,h^f=hk,i^c=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72237948, #72237974, #72238670, …
maximal normal subgroup of#72237948, #72237974, #72238670, …

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.5);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);