#145729

order	512
nilpotency class	4
isoclinism	isoclinic class of #145727
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

341 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,5)(4,6)(7,8)(9,10,18,26)(11,27,32,40)(12,33,29,38)(13,22,34,23)(14,21,15,36)(16,25,17,24)(19,35,39,28)(20,31,37,30);(1,3)(2,5)(4,7)(6,8)(9,11,18,32)(10,19,26,39)(12,34,29,13)(14,28,15,35)(16,31,17,30)(20,21,37,36)(22,27,23,40)(24,33,25,38);(5,8)(9,12,13,11,17,28,15,31,18,29,34,32,16,35,14,30)(10,20,21,39,25,40,23,38,26,37,36,19,24,27,22,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52925);