#489239

order	512
nilpotency class	4
isoclinism	isoclinic class of #488912
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

711 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 711 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

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, 396435);