#488993

order	512
nilpotency class	4
isoclinism	isoclinic class of #488889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

766 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396189);