#490532

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	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 458 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)(17,18,20,25)(19,28,29,23)(21,24,32,39)(22,35,33,27)(26,41,40,34)(30,42,45,36)(31,38,46,44)(37,47,43,48);(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)(32,45)(33,46)(34,47)(35,44)(39,42)(40,43)(41,48);(18,24)(21,34)(22,33)(23,36)(25,39)(26,27)(28,42)(30,47)(31,46)(32,41)(35,40)(37,38)(43,44)(45,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(19,29)(23,28)(30,45)(31,46)(36,42)(37,43)(38,44)(47,48)

Polycyclic presentation:

2^9|a^2=i,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, 397728);