#507614

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

451 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 168 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=g,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414810);