#144922

order	512
nilpotency class	4
isoclinism	isoclinic class of #144895
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	C₂³×C₈	C₂³
prime radical	#30253	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

437 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=dg,e^2=gi,f^2=gi,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=fg,g^b=gi,h^b=hi,d^c=di,e^c=egi,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52118);