#417436

order	512
nilpotency class	3
isoclinism	isoclinic class of #394183
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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

912 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 242 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)(17,18,20,22)(19,21,23,24)(25,26)(27,33)(28,37)(29,31)(30,36)(32,34)(35,39)(38,40);(9,10)(11,16)(12,14)(13,15)(17,19,20,23)(18,21,22,24)(25,27,34,40,29,35,28,36)(26,30,37,39,31,38,32,33);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(18,22)(19,23)(27,35)(30,38)(33,39)(36,40);(9,12,13,11)(10,14,15,16)(18,22)(21,24)(26,31)(27,35)(32,37)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=fgi,d^2=gh,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=di

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, 324632);