#6407898

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
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	#2819	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1380 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,21,26,22,24,30,32)(19,23,28,31,27,20,25,29);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,24)(21,28)(22,27)(25,30)(26,31)(29,32);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,24)(21,28)(22,27)(25,30)(26,31)(29,32);(2,6)(5,8)(10,13)(14,16)(18,24)(19,27)(25,28)(26,32);(17,20,21,29,22,23,30,31)(18,25,26,19,24,28,32,27)

Polycyclic presentation:

2^9|a^2=h,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=cfg,d^b=dfi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6315094);