#72322247

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	#30400
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#13323	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20383 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1963 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:

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

Polycyclic presentation:

2^10|b^2=e,c^2=efj,d^2=i,e^2=j,h^2=ij,i^2=j,b^a=bh,c^a=cg,c^b=cf,d^a=dg,d^b=dg,d^c=deij,e^a=ej,g^c=gj,g^d=gj,h^a=hi,h^b=hij,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322256, #72322591
maximal normal subgroup of	#72322256, #72322591

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