#145452

order	512
nilpotency class	4
isoclinism	isoclinic class of #145363
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

287 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

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

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, 52648);