#72322418

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21526
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1915 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=efh,d^2=i,e^2=j,f^2=i,g^2=ij,h^2=ij,i^2=j,b^a=bf,c^a=cfi,c^b=ceij,d^b=defgij,d^c=defgij,e^a=ej,e^c=ej,e^d=ej,f^b=fi,f^c=fi,f^d=fj,g^b=gij,g^c=gij,g^d=gj,h^b=hij,h^c=hi,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322428, #72322569, #72322623, …
maximal normal subgroup of	#72322428, #72322569, #72322623, …

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