#491070

order	512
nilpotency class	4
isoclinism	isoclinic class of #491025
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 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

410 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=e,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=dgi,d^c=dg,f^c=fh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398266);