#6376667

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

1492 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)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18,22,25,21,24,26,19)(20,28,30,27,29,23,31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19,26,24,21,25,22,18)(20,27,31,28,29,32,30,23);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,20,21,29)(18,23,24,28)(19,27,25,32)(22,30,26,31);(1,3,4,7)(2,6,5,8);(20,29)(23,28)(27,32)(30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,e^c=ei

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, 6283863);