#491187

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

403 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,11,13,16)(4,9,5,8)(7,14,15,12)(18,24)(19,30)(20,22)(21,25)(26,41)(27,38)(28,37)(29,39)(31,33)(32,46)(34,35)(36,43)(42,48)(44,47);(17,18,25,37,20,27,35,39,23,29,34,38,22,28,21,24)(19,26,40,41,30,43,46,48,33,44,45,47,31,42,32,36);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19)(18,26)(20,30)(21,32)(22,31)(23,33)(24,36)(25,40)(27,43)(28,42)(29,44)(34,45)(35,46)(37,41)(38,47)(39,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(17,20,23,22)(18,27,29,28)(19,31,33,30)(21,25,35,34)(24,37,39,38)(26,42,44,43)(32,45,46,40)(36,47,48,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=eg,d^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,d^c=di,f^c=fh

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, 398383);