#325937

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egh,c^2=fh,d^2=ehi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,g^c=gi

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, 233133);