#72322234

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32659
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#135	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1163 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=efg,d^2=gj,e^2=j,h^2=i,i^2=j,b^a=bhij,c^a=cf,c^b=cfg,d^a=dfh,d^b=dfj,d^c=defgij,e^a=ej,f^c=fj,g^c=gj,h^b=hi,h^d=hij,i^b=ij,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322256, #72322386, #72322919
maximal normal subgroup of	#72322256, #72322386, #72322919

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