#6908169

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×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6815365);