#145949

order	512
nilpotency class	4
isoclinism	isoclinic class of #145934
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	#468
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

231 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

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, 53145);