#72307454

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#3184
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂²	#23573
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4730 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:

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

Polycyclic presentation:

2^10|b^2=fj,c^2=fj,d^2=egij,e^2=j,g^2=i,i^2=j,b^a=bf,d^a=dh,d^b=dg,d^c=de,e^b=ej,f^d=fj,g^b=gi,g^c=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307537, #72307810
maximal normal subgroup of	#72307537, #72307810

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