#6385593

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	#2714	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1561 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)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,25,22,27,20,30)(19,24,28,31,29,21,26,32);(1,2,4,5)(3,7,8,6)(9,11,12,15)(10,13,14,16)(17,19,22,29)(18,21,27,24)(20,26,23,28)(25,32,30,31);(10,14)(13,16)(17,19)(18,24)(20,26)(21,27)(22,29)(23,28)(25,31)(30,32);(1,3,4,8)(2,6,5,7)(17,20,22,23)(18,25,27,30)(19,28,29,26)(21,31,24,32);(17,21,23,32,22,24,20,31)(18,26,25,19,27,28,30,29)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,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, 6292789);