#490530

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
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 449 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) = 256

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,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, 397726);