#72307293

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#8375
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂	C₂×#8375
prime radical	#10576101	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

5836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=efij,d^2=hi,e^2=ij,f^2=i,h^2=j,i^2=j,b^a=be,c^a=cj,c^b=cj,d^a=dhi,d^b=defgij,e^a=ei,e^b=eh,f^a=fh,f^b=fh,f^d=fhi,f^e=fj,g^a=ghi,g^b=ghij,g^d=ghi,g^e=gj,h^a=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307300, #72307601
maximal normal subgroup of	#72307300, #72307601

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