#72319741

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄²×D₄
commutator	C₄²	C₂²×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂²	D₄×#97
prime radical	C₂×#135	M₁₆
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #11404 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=e,c^2=efhij,d^2=g,e^2=h,f^2=hi,g^2=i,h^2=j,c^b=chi,d^a=dg,d^b=di,d^c=di,e^c=ej,f^b=fj,f^c=fj,g^a=gi

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319854, #72320132
maximal normal subgroup of	#72319854, #72320132

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