#484217

order	512
nilpotency class	4
isoclinism	isoclinic class of #484217
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×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1166 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 140 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 — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 391413);