#6405161

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
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

1426 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)(3,6)(4,8)(5,7)(9,10,13,16)(11,15,12,14)(17,18,22,24,21,23,19,25)(20,26,29,31,28,30,27,32);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,15)(13,16)(18,23)(24,25)(26,30)(31,32);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,19,21,22)(18,24,23,25)(20,27,28,29)(26,31,30,32);(9,11,13,12)(10,15,16,14)(17,19,21,22)(18,25,23,24)(20,27,28,29)(26,32,30,31);(9,12,13,11)(10,15,16,14)(17,20,22,29,21,28,19,27)(18,26,24,31,23,30,25,32)

Polycyclic presentation:

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

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, 6312357);