#144686

order	512
nilpotency class	4
isoclinism	isoclinic class of #144686
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#29748	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 425 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)(10,16)(11,20)(12,14)(13,26)(17,32)(18,35)(19,24)(21,39)(22,27)(23,38)(25,28)(29,37)(30,33)(34,36);(1,3)(2,5)(4,7)(6,8)(9,10)(11,21)(12,24)(13,18)(14,16)(15,19)(17,23)(20,36)(22,30)(25,40)(26,31)(27,39)(28,35)(29,34)(32,37)(33,38);(5,8)(9,11,12,22,15,23,14,29)(10,17,16,30,19,21,24,34)(13,27,25,20,28,37,26,38)(18,32,31,36,35,39,40,33)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51882);