#353096

order	512
nilpotency class	3
isoclinism	isoclinic class of #352081
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

582 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

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, 260292);