#490772

order	512
nilpotency class	4
isoclinism	isoclinic class of #490699
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

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

normal subgroups only

This group has 433 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)(7,13)(11,16)(12,15)(17,18,24,28)(19,21,26,22)(20,31,36,46)(23,39,41,30)(25,42,43,35)(27,37,45,38)(29,40,32,48)(33,47,34,44);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,19,22,18,24,26,21,28)(20,32,34,43,36,29,33,25)(23,30,38,27,41,39,37,45)(31,35,47,48,46,42,44,40);(3,11)(7,15)(10,16)(12,13)(17,20,24,36)(18,25,28,43)(19,29,26,32)(21,33,22,34)(23,40,41,48)(27,44,45,47)(30,46,39,31)(35,37,42,38);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(17,21,24,22)(18,26,28,19)(20,33,36,34)(23,37,41,38)(25,32,43,29)(27,39,45,30)(31,47,46,44)(35,40,42,48)

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,c^b=cg,d^b=di,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, 397968);