#144717

order	512
nilpotency class	4
isoclinism	isoclinic class of #144716
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

430 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 66 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)(9,10,12,17)(11,20,25,39)(13,27,26,40)(14,19,15,32)(16,33,34,22)(18,30,28,31)(21,37,36,29)(23,38,24,35);(1,3)(2,5)(4,7)(6,8)(9,10)(11,21)(12,17)(13,28)(14,32)(15,19)(16,23)(18,26)(20,37)(22,35)(24,34)(25,36)(27,30)(29,39)(31,40)(33,38);(5,8)(9,11,14,24,12,25,15,23)(10,16,32,21,17,34,19,36)(13,29,30,22,26,37,31,33)(18,35,40,20,28,38,27,39)

Polycyclic presentation:

2^9|a^2=i,c^2=dg,e^2=gi,f^2=i,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,f^c=fi,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, 51913);