#507705

order	512
nilpotency class	4
isoclinism	isoclinic class of #501524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

499 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=ehi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,f^b=fi,e^c=ei,f^c=fh,h^c=hi

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, 414901);