#145982

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:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=dg,c^2=gh,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=eg,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, 53178);