#72322437

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	#32002
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20383 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1241 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=efg,d^2=ij,e^2=j,h^2=i,i^2=j,b^a=bhij,c^a=cf,c^b=cj,d^b=dg,d^c=deij,e^a=ej,f^c=fj,f^d=fj,g^c=gj,g^d=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322466, #72322569, #72322883
maximal normal subgroup of	#72322466, #72322569, #72322883

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