#352409

order	512
nilpotency class	3
isoclinism	isoclinic class of #347638
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

616 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)(10,13)(11,14)(15,16)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,31,30,27);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16)(18,22)(23,30)(24,29)(25,32);(17,19,21,27,20,26,28,31)(18,23,24,25,22,30,29,32);(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=f,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,g^d=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259605);