#145956

order	512
nilpotency class	4
isoclinism	isoclinic class of #143938
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2065
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 259 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:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=d,d^2=g,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=fg,g^a=gi,h^a=hi,c^b=cf,e^b=ei,d^c=dg,e^c=eh,f^c=fg,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53152);