#72313657

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#8376
commutator	C₂²×C₈	C₂×C₄²
Frattini, cosocle	C₂²×#131	C₂³
prime radical	C₂×#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

1793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=hj,b^2=g,c^2=d,e^2=j,f^2=i,g^2=ij,i^2=j,b^a=be,c^a=chj,c^b=cf,d^b=dj,e^b=eh,e^c=ej,f^b=fj,f^c=fi,g^a=ghj,g^c=gi,g^e=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313675, #72313820, #72314056
maximal normal subgroup of	#72313675, #72313820, #72314056

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