#6661070

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660376
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
prime radical	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^c=dfh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6568266);