#7506819

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1649 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7414015);