#440863

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348059);