#483289

order	512
nilpotency class	4
isoclinism	isoclinic class of #482975
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

673 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 673 conjugacy classes of subgroups — too many to draw. Only its 240 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=ci,d^a=di,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=dgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390485);