#72322293

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32554
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#135	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20438 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(6,10)(7,13,18,20)(8,22,24,14)(9,11)(15,16)(17,23)(19,27,21,30)(25,32,26,28)(29,34)(31,35);(1,2,4,3)(5,6,11,15,12,16,9,10)(7,13,19,30,18,20,21,27)(8,14,25,32,24,22,26,28)(17,31,36,35,23,29,33,34);(1,3,4,2)(5,7,9,21,12,18,11,19)(6,13,10,27,16,20,15,30)(8,23,26,36,24,17,25,33)(14,31,28,34,22,29,32,35);(2,3)(5,8)(6,14)(7,17)(9,26)(10,28)(11,25)(12,24)(13,29)(15,32)(16,22)(18,23)(19,36)(20,31)(21,33)(27,35)(30,34)

Polycyclic presentation:

2^10|a^2=e,b^2=efj,c^2=ef,f^2=j,g^2=i,h^2=i,i^2=j,b^a=bij,c^a=cg,c^b=ceghij,d^a=dh,d^b=dfgh,d^c=dfij,e^c=ej,e^d=ej,f^a=fj,f^c=fj,f^d=fj,g^a=gi,g^b=gj,g^c=gi,g^d=gij,h^a=hi,h^b=hj,h^c=hij,h^d=hij,i^a=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322302, #72322676
maximal normal subgroup of	#72322302, #72322676

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