#353058

order	512
nilpotency class	3
isoclinism	isoclinic class of #352019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,20)(13,15)(14,19)(16,21)(18,23)(22,24)(25,26,28,30)(27,33,34,40)(29,37,36,32)(31,39,38,35);(1,3,4,8)(2,5,6,7)(11,18)(14,22)(17,23)(19,24)(26,30)(31,38)(32,37)(33,40);(9,11,12,19,13,18,21,24)(10,14,16,17,15,22,20,23);(10,15)(14,22)(16,20)(17,23)(25,27,29,35,28,34,36,39)(26,31,32,33,30,38,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260254);