#6468972

order	512
nilpotency class	3
isoclinism	isoclinic class of #23918
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2020 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18)(19,26)(20,25)(21,29)(22,24)(23,27)(28,32)(30,31);(9,10)(11,16)(12,14)(13,15)(17,19,22,28)(18,23,24,31)(20,27,21,30)(25,32,29,26);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,24)(23,31)(25,29)(26,32);(10,14)(13,16)(17,20,22,21)(18,25,24,29)(19,27,28,30)(23,32,31,26);(1,4,5,3)(2,6,7,8)

Polycyclic presentation:

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

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, 6376168);