#72343644

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂³	C₂²×C₄×D₄
prime radical	C₂×#3280	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #21101 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3279 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=g,e^2=hj,f^2=i,h^2=j,b^a=bg,c^a=cgij,c^b=cgij,d^b=dgij,e^a=ej,e^c=eh,h^c=hj

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343649, #72343660, #72343895
maximal normal subgroup of	#72343649, #72343660, #72343895

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