#507403

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

500 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,26,34,41,46,45,21,25,33,40,23,27,36,43)(19,20,29,31,44,37,48,47,28,24,38,39,30,32,42,35);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,25)(21,28)(22,29)(23,30)(26,39)(27,37)(31,40)(32,41)(33,38)(34,44)(35,45)(36,42)(43,47)(46,48);(11,15)(13,16)(17,20,21,24)(18,19,25,28)(22,35,33,47)(23,37,34,32)(26,42,40,48)(27,44,41,30)(29,45,38,43)(31,46,39,36);(1,3,4,8)(2,6,5,7)(18,25)(19,28)(26,40)(27,41)(29,38)(30,44)(42,48)(43,45)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,h^c=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, 414599);