#6405214

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

1330 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)(17,18,24,28,23,22,21,27)(19,29,20,26,30,25,32,31);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,19,23,30)(18,25,22,29)(20,21,32,24)(26,28,31,27);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20)(18,26)(19,21)(22,31)(23,32)(24,30)(25,28)(27,29);(9,11,13,12)(10,15,16,14)(17,21,23,24)(18,27,22,28)(19,20,30,32)(25,31,29,26);(9,12,13,11)(10,15,16,14)(17,22,24,27,23,18,21,28)(19,25,20,31,30,29,32,26)

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6312410);