#72322281

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#31064
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#498831	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20383 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1956 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)(5,6,11,16,9,15,26,32,12,14,29,28,10,8,24,25)(7,17,22,34,21,13,30,31,23,19,27,36,20,18,35,33);(2,4)(5,7)(6,13)(8,17)(9,21)(10,20)(11,27)(12,23)(14,18)(15,19)(16,33)(22,29)(24,30)(25,36)(26,35)(28,31)(32,34);(1,2)(3,4)(5,6)(7,18)(8,9)(10,15)(11,25)(12,14)(13,23)(16,24)(17,20)(19,21)(22,36)(26,28)(27,34)(29,32)(30,31)(33,35);(2,4)(5,8)(6,10)(7,19,23,17)(9,14)(11,28)(12,15)(13,20,18,21)(16,29)(22,31,27,33)(24,25)(26,32)(30,34,35,36)

Polycyclic presentation:

2^10|a^2=h,d^2=gij,f^2=j,g^2=j,h^2=ij,i^2=j,b^a=bf,c^a=ch,c^b=cfgj,d^a=dfhi,d^b=de,d^c=dfij,e^a=ej,e^c=ej,e^d=ej,f^b=fj,f^c=fj,f^d=fj,g^b=gj,g^c=gj,g^d=gj,h^b=hj,h^c=hi,h^d=hi,i^c=ij,i^d=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322302, #72322569, #72323029
maximal normal subgroup of	#72322302, #72322569, #72323029

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