#349345

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
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

1104 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)(19,24)(20,29)(21,26)(22,31)(23,27)(25,30)(28,32);(1,3,4,8)(2,5,6,7)(17,19,23,20)(18,24,27,29)(21,30,22,28)(25,31,32,26);(9,10)(11,13)(12,14)(15,16)(17,20,23,19)(18,25,27,32)(21,30,22,28)(24,31,29,26);(9,11,12,15)(10,13,14,16)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,32,29,25)

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=h,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh,g^c=gi,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, 256541);