#476076

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

956 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 250 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=gi,c^2=fi,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de

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, 383272);