#72307115

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#6453969
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #60207 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4761 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=g,c^2=g,e^2=fh,h^2=i,i^2=j,b^a=bj,d^b=dgj,d^c=dgj,e^a=eh,e^b=ef,e^c=efi,e^d=egj,f^d=fj,g^e=gj,h^a=hi,h^c=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307129, #72307752, #72307980, …
maximal normal subgroup of	#72307129, #72307752, #72307980, …
maximal quotient of	C₂×#72307115

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