#349216

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×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

990 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)(10,13)(11,14)(15,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16);(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(9,10,12,13)(11,14,15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31)

Polycyclic presentation:

2^9|a^2=ehi,b^2=e,d^2=h,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, 256412);