#353092

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

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

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

normal subgroups only

This group has 590 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)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,23)(22,24)(25,26)(27,32)(28,37)(29,39)(30,34)(31,36)(33,38)(35,40);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,21)(23,24)(27,35)(28,36)(31,37)(32,40);(9,11,12,18,13,19,21,24)(10,14,16,17,15,22,20,23)(25,27,29,36,30,35,38,28)(26,31,33,32,34,37,39,40);(10,15)(14,22)(16,20)(17,23)(25,28,38,35,30,36,29,27)(26,32,39,31,34,40,33,37)

Polycyclic presentation:

2^9|a^2=e,b^2=e,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=ci,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, 260288);