#72322792

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	#24089
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#6442941	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2932 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2254 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=i,d^2=efgij,e^2=j,f^2=j,h^2=i,i^2=j,b^a=bhij,c^a=cf,c^b=cfi,d^a=df,d^b=dg,d^c=defg,e^a=ej,f^b=fj,f^c=fj,g^c=gj,h^b=hi,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	#72322804, #72322854, #72322945
maximal normal subgroup of	#72322804, #72322854, #72322945

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