#72257596

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32468
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#525	D₄
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1746 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:

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

Polycyclic presentation:

2^10|b^2=e,c^2=h,d^2=fgi,e^2=i,i^2=j,b^a=bf,c^b=ceij,d^a=dfg,d^b=dhi,d^c=dfh,e^c=ei,f^d=fj,g^b=gj,g^d=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257601, #72257632, #72257680
maximal normal subgroup of	#72257601, #72257632, #72257680

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