#145975

order	512
nilpotency class	4
isoclinism	isoclinic class of #145951
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	#460
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

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

normal subgroups only

This group has 218 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:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=dg,c^2=g,d^2=g,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgi,c^b=cf,f^b=fi,d^c=dgi,e^c=eh,f^c=fg,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, 53171);