Finite Group Databasebeta

← Prev (#72283581) Next (#72283583) →

#72283582

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#150393
commutator#379C22×C4
Frattini, cosocle#1781C23
prime radicalC22×H(5,𝔽2)D4
LCSG ▷ #379 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1101 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 1101 conjugacy classes of subgroups — too many to draw. Only its 55 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=fh,d^2=g,e^2=ghi,b^a=be,c^a=cfhij,c^b=cfhi,d^b=df,d^c=dhij,e^a=efghi,e^b=egj,e^c=ehij,e^d=ehij,f^a=fh,f^b=fij,f^d=fi,f^e=fij,g^b=gi,g^c=gj,g^e=gj,h^a=hi,h^b=hij,h^d=hj,h^e=hj,i^a=ij

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283653, #72283916, #72285283, …
maximal normal subgroup of#72283653, #72283916, #72285283, …

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