#380733

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1270 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 338 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)(3,4)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,26,30,32,29,33,28,34)(27,31,37,40,35,39,36,38);(5,6,7,8)(10,14)(15,21)(16,22)(23,24)(25,27)(26,31)(28,36)(29,35)(30,37)(32,40)(33,39)(34,38);(2,4)(6,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(26,32)(27,35)(28,30)(31,38)(33,34)(39,40);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(26,33)(31,39)(32,34)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287929);