#72321501

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23960
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#411663	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2932 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=e,c^2=i,d^2=egh,e^2=j,f^2=i,g^2=i,h^2=ij,i^2=j,b^a=bh,c^a=cefgij,c^b=cefg,d^a=dghi,d^b=dgh,d^c=deghi,e^a=ej,e^c=ej,f^b=fi,f^d=fj,g^b=gi,g^d=gj,h^b=hij,h^c=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321614, #72321803
maximal normal subgroup of	#72321614, #72321803

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