#417409

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

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

normal subgroups only

This group has 960 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)(19,21)(20,22)(23,24)(25,26,29,31)(27,33,35,40)(28,37,34,32)(30,38,39,36);(9,10)(11,14)(12,13)(15,16)(17,19,20,23)(18,21,22,24)(25,27,34,38,29,35,28,36)(26,30,37,40,31,39,32,33);(1,3,4,8)(2,5,6,7)(10,13)(14,16)(19,23)(21,24)(26,31)(30,39)(32,37)(33,40);(9,11,12,15)(10,14,13,16)(18,22)(21,24)(26,31)(27,35)(32,37)(36,38)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=fi,d^2=ghi,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, 324605);