#6757345

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 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)(11,14)(12,13)(15,16)(17,18,22,25,21,27,20,26)(19,23,28,24,30,32,29,31);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,21,30)(18,23,27,32)(20,29,22,28)(24,26,31,25);(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,25)(26,32)(27,31);(2,6)(5,7)(10,13)(14,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,25)(26,31)(27,32);(3,8)(5,7)(17,20,21,22)(18,25,27,26)(19,28,30,29)(23,31,32,24)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6664541);