#6908562

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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

1774 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6815758);