#145033

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

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

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

normal subgroups only

This group has 404 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)(7,8)(9,10,15,20)(11,21,23,38)(12,26,29,18)(13,30,27,17)(14,31,28,19)(16,33,35,24)(22,36,40,37)(25,34,32,39);(2,5)(6,7)(11,22)(12,27)(13,29)(14,28)(16,34)(17,26)(18,30)(19,31)(21,37)(23,40)(24,32)(25,33)(35,39)(36,38);(2,6)(4,8)(5,7)(9,11,14,32,15,23,28,25)(10,16,19,37,20,35,31,36)(12,22,27,33,29,40,13,24)(17,34,26,38,30,39,18,21)

Polycyclic presentation:

2^9|a^2=i,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, 52229);