#6407321

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, prime radical	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6314517);