#6385678

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1404 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ci,d^b=df,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6292874);