#490764

order	512
nilpotency class	4
isoclinism	isoclinic class of #490693
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 ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 436 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)(7,13)(11,16)(12,15)(17,18,21,24)(19,28,30,45)(20,26,23,38)(22,35,34,46)(25,41,39,31)(27,33,40,37)(29,44,32,48)(36,43,47,42);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(18,24)(25,39)(26,38)(27,40)(28,45)(35,46)(42,43)(44,48);(3,11)(7,15)(10,16)(12,13)(17,19,21,30)(18,25,24,39)(20,32,23,29)(22,36,34,47)(26,43,38,42)(27,44,40,48)(28,46,45,35)(31,33,41,37);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(17,20,21,23)(18,26,24,38)(19,29,30,32)(22,33,34,37)(25,42,39,43)(27,35,40,46)(28,44,45,48)(31,47,41,36)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=i,d^2=gi,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^c=di,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397960);