#145996

order	512
nilpotency class	4
isoclinism	isoclinic class of #145988
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2067
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#478
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

201 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53192);