#144454

order	512
nilpotency class	4
isoclinism	isoclinic class of #144454
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2023
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 509 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)(11,19,21,36)(12,23)(13,27)(15,29,20,38)(16,31)(18,33)(22,25,28,37)(24,39)(26,35)(30,32,34,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,20,21,15)(12,24)(13,18)(14,17)(16,26)(19,29,36,38)(22,34,28,30)(23,39)(25,32,37,40)(27,33)(31,35);(5,8)(9,11)(10,15)(12,25)(13,28)(14,21)(16,32)(17,20)(18,34)(19,35)(22,27)(23,36)(24,40)(26,37)(29,39)(30,33)(31,38)

Polycyclic presentation:

2^9|a^2=d,b^2=d,e^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,f^b=fi,d^c=di,e^c=egi,g^c=gi

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, 51650);