#6464480

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1632 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)(4,5)(7,8)(17,18,20,23,21,24,29,32)(19,25,26,30,27,31,28,22);(9,10)(11,15)(12,13)(14,16)(17,19,21,27)(18,22,24,30)(20,28,29,26)(23,31,32,25);(2,5)(6,8)(10,13)(14,15);(9,11,12,16)(10,14,13,15);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi

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, 6371676);