#72321815

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#31064
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2225 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,h^2=i,i^2=j,b^a=bhij,c^a=cg,c^b=cef,d^a=dh,d^b=dg,d^c=dfg,e^a=ej,e^d=ej,g^b=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	#72321841, #72322691, #72323029
maximal normal subgroup of	#72321841, #72322691, #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, 2, F.5);;
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.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 7, 2, 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);