#440766

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1185 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)(6,8)(9,10,12,14)(11,13,15,16)(17,18,21,25,22,26,30,29)(19,23,27,31,20,24,28,32);(1,2,4,5)(3,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31);(2,5)(6,8)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)

Polycyclic presentation:

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

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, 347962);