#512370

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

814 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419566);