#507652

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

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

normal subgroups only

This group has 434 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)(17,18,20,25,32,40,44,39,22,27,33,38,21,26,31,24)(19,23,29,36,42,47,48,45,28,35,41,46,30,37,43,34);(2,5)(6,7)(9,10,12,14)(11,15,16,13)(17,19)(18,23)(20,29)(21,30)(22,28)(24,34)(25,36)(26,37)(27,35)(31,43)(32,42)(33,41)(38,46)(39,45)(40,47)(44,48);(9,11,12,16)(10,13,14,15)(18,24)(20,31)(21,32)(23,34)(25,26)(27,39)(29,43)(30,42)(33,44)(35,45)(36,37)(38,40)(41,48)(46,47);(1,3,4,8)(2,6,5,7)(19,28)(23,35)(29,41)(30,42)(34,45)(36,46)(37,47)(43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=e,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, 414848);