#484637

order	512
nilpotency class	4
isoclinism	isoclinic class of #484212
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

933 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 933 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 391833);