#349207

order	512
nilpotency class	3
isoclinism	isoclinic class of #346294
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1154 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)(3,7,8,5)(13,16)(17,18,21,23)(19,26,28,32)(20,25,22,29)(24,30,31,27);(1,3,4,8)(2,5,6,7)(18,23)(24,31)(25,29)(26,32);(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,27)(21,28)(22,30)(23,31)(25,32)(26,29);(9,11,12,15)(10,13,14,16)(17,20,21,22)(18,25,23,29)(19,27,28,30)(24,32,31,26)

Polycyclic presentation:

2^9|a^2=e,b^2=eh,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,f^c=fh,g^c=gi,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, 256403);