#491003

order	512
nilpotency class	4
isoclinism	isoclinic class of #490891
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

421 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 126 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)(20,31,34,46)(23,38,40,30)(25,41,43,33)(27,36,45,37)(29,39,42,48)(32,44,35,47);(1,3,4,11,6,13,5,12)(2,7,14,16,9,15,8,10)(17,19,22,18,24,26,21,28)(20,29,32,25,34,42,35,43)(23,30,37,27,40,38,36,45)(31,41,47,39,46,33,44,48);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(17,20,24,34)(18,25,28,43)(19,29,26,42)(21,35,22,32)(23,39,40,48)(27,44,45,47)(30,46,38,31)(33,36,41,37);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,21,24,22)(18,26,28,19)(20,32,34,35)(23,36,40,37)(25,29,43,42)(27,38,45,30)(31,44,46,47)(33,48,41,39)

Polycyclic presentation:

2^9|a^2=i,b^2=egi,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,d^c=dg,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398199);