#72285488

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#245270	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=f,c^2=gj,e^2=gj,b^a=bfh,c^a=ch,c^b=cfg,d^a=dfij,d^b=dfh,d^c=dfhij,e^a=efgij,e^b=efh,e^c=egj,e^d=ehij,f^b=fi,f^c=fi,f^e=fij,g^a=gij,g^b=gi,g^d=gj,h^a=hi,h^b=hi,h^c=hij,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285599
maximal normal subgroup of	#72285599

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