#6385681

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1376 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)(17,18,24,26,23,27,21,31)(19,28,29,32,20,25,30,22);(1,2,5,6)(3,7,4,8)(9,11,12,15)(10,13,14,16)(17,19,23,20)(18,25,27,28)(21,30,24,29)(22,31,32,26);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,20,23,19)(18,25,27,28)(21,29,24,30)(22,31,32,26);(1,4,5,3)(2,7,6,8)(17,21,23,24)(18,26,27,31)(19,29,20,30)(22,25,32,28);(17,22,24,28,23,32,21,25)(18,19,26,29,27,20,31,30)

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,d^b=df,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6292877);