#467150

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
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, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1280 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374346);