#488749

order	512
nilpotency class	4
isoclinism	isoclinic class of #488707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

789 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 789 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395945);