#144522

order	512
nilpotency class	4
isoclinism	isoclinic class of #144521
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	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
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

345 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=di,e^2=g,f^2=hi,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=fh,d^c=di,e^c=egi,f^c=fhi,g^c=gi

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, 51718);