#72322809

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	C₂×C₄	C₂×#469
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂²	C₂×#2076
prime radical	C₂³×D₄	Q₁₆
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂²×C₈×D₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1991 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:

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

Polycyclic presentation:

2^10|a^2=h,c^2=g,d^2=efg,g^2=j,h^2=ij,i^2=j,c^a=ch,d^a=dghij,d^b=df,d^c=de,e^a=ej,h^c=hi,h^d=hi,i^c=ij,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322844, #72322941
maximal normal subgroup of	#72322844, #72322941

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