#72318160

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#12687
commutator	C₂³×Q₈	C₂²×C₄
Frattini, cosocle	C₂²×#124	C₂³
prime radical	C₂×#23915	C₂
LCS	G ▷ C₂³×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂²×#535 ◁ G
LECS	G ▷ C₂²×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4035 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=d,c^2=egj,e^2=j,g^2=j,h^2=j,b^a=bf,c^a=ce,c^b=cegj,d^c=dh,e^a=eh,e^b=ej,e^c=ej,e^d=eij,f^c=fj,g^a=gh,g^b=gj,g^c=gj,g^d=gij,h^a=hi,h^c=hj,h^d=hj,h^e=hj,h^g=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318162, #72318179, #72318461
maximal normal subgroup of	#72318162, #72318179, #72318461

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