#507656

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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

430 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 166 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)(17,18,21,26,33,41,47,39,20,25,32,40,22,27,34,24)(19,28,30,43,45,38,48,36,29,23,35,37,31,44,46,42);(2,5)(6,7)(9,10,12,14)(11,15,16,13)(17,19,20,29)(18,23,25,28)(21,30,32,35)(22,31,33,45)(24,36,39,42)(26,37,40,43)(27,38,41,44)(34,46,47,48);(9,11,12,16)(10,13,14,15)(18,24)(21,34)(22,33)(23,36)(25,39)(26,27)(28,42)(30,46)(31,45)(32,47)(35,48)(37,38)(40,41)(43,44);(1,3,4,8)(2,6,5,7)(18,25)(19,29)(24,39)(26,40)(27,41)(30,35)(31,45)(46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=g,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, 414852);