#6385215

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6292411);