#6856877

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667652
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	C₂×#459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1462 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)(3,6)(4,8)(5,7)(9,10,13,15)(11,14,12,16)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,31)(30,32);(1,3,6,2)(4,8,5,7)(9,10)(11,16)(12,14)(13,15)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,30,25,31);(9,11,13,12)(10,14,15,16)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(1,4,6,5)(2,7,3,8)(9,12,13,11)(10,14,15,16)(18,24)(22,26)(23,25)(30,31);(1,5,6,4)(2,7,3,8)(19,27)(22,26)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=h,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6764073);