#145108

order	512
nilpotency class	4
isoclinism	isoclinic class of #145011
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
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 ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

344 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 80 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)(4,6,8,7)(9,10,15,20)(11,21,23,38)(12,18,28,39)(13,17,30,37)(14,31,27,19)(16,29,34,24)(22,40,26,35)(25,36,32,33);(2,5)(6,7)(11,22)(12,13)(14,27)(16,33)(17,18)(19,31)(21,35)(23,26)(24,25)(28,30)(29,32)(34,36)(37,39)(38,40);(2,6)(4,8)(5,7)(9,11,14,32,15,23,27,25)(10,16,19,40,20,34,31,35)(12,26,13,29,28,22,30,24)(17,36,18,38,37,33,39,21)

Polycyclic presentation:

2^9|a^2=gh,c^2=dg,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^c=egi,f^c=fh,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, 52304);