#490526

order	512
nilpotency class	4
isoclinism	isoclinic class of #490434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

492 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=ei,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,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, 397722);