#72250898

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	#23708
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×D₈	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1476 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=ehj,c^2=hi,d^2=fh,e^2=i,g^2=j,h^2=j,i^2=j,b^a=bij,c^a=cegj,c^b=cegj,d^a=df,d^b=dhij,d^c=dfgh,e^b=ej,e^c=ei,e^d=ej,f^a=fj,f^b=fj,g^c=gj,g^d=gj,h^a=hj,h^b=hj,h^c=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250937, #72257186, #72257613
maximal normal subgroup of	#72250937, #72257186, #72257613

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