#145277

order	512
nilpotency class	4
isoclinism	isoclinic class of #145276
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=dh,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,d^c=di,e^c=eghi,f^c=fg,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52473);