#72240850

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2085
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁵	#135
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1671 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);(1,2,4,3)(5,6,10,16,11,12,9,15)(7,17,25,28,19,13,20,31)(8,21,22,29,24,14,23,30)(18,27,34,35,32,26,33,36);(2,3)(6,12)(7,18,19,32)(8,22)(9,10)(13,26,17,27)(14,29)(20,34,25,33)(21,30)(23,24)(28,36,31,35);(1,3,4,2)(5,7,8,18,10,20,22,33,11,19,24,32,9,25,23,34)(6,13,21,26,16,28,29,35,12,17,14,27,15,31,30,36)

Polycyclic presentation:

2^10|b^2=fhj,c^2=hi,d^2=eghi,e^2=i,h^2=j,i^2=j,b^a=bf,c^b=cfg,d^a=df,d^b=dghi,d^c=de,e^b=ej,e^c=ei,g^b=gj,g^d=gj,h^c=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240864, #72250220, #72257671
maximal normal subgroup of	#72240864, #72250220, #72257671

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