#72320737

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	#24089
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3050	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1882 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,d^2=fg,e^2=j,h^2=i,i^2=j,b^a=bhij,c^a=cf,c^b=cg,d^b=dfg,d^c=defgi,e^a=ej,f^d=fj,g^d=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320850, #72322868, #72322913
maximal normal subgroup of	#72320850, #72322868, #72322913

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