#72322266

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	#32468
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

1121 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=efgi,d^2=fgij,e^2=j,f^2=i,g^2=i,h^2=i,i^2=j,b^a=bhij,c^a=cf,c^b=cfgij,d^a=dfhij,d^b=dfhi,d^c=deghj,e^a=ej,f^b=fi,f^c=fi,g^b=gi,g^c=gi,h^b=hi,h^c=hij,i^b=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322302, #72322354, #72322844
maximal normal subgroup of	#72322302, #72322354, #72322844

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