#490502

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	C₂³	#459
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=eg,c^2=i,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, 397698);