#72240845

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23764
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#472	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,d^2=efgi,e^2=i,f^2=j,i^2=j,b^a=bg,c^b=cfghij,d^a=de,d^b=dgh,d^c=defhj,e^a=ei,e^b=ej,e^c=eij,f^a=fj,f^c=fj,f^d=fj,g^b=gj,g^d=gj,h^d=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240864, #72257268, #72257552
maximal normal subgroup of	#72240864, #72257268, #72257552

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