#146002

order	512
nilpotency class	4
isoclinism	isoclinic class of #146000
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2065
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

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

normal subgroups only

This group has 189 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=deh,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=fgi,c^b=cf,f^b=fg,d^c=dgi,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53198);