#507404

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	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 485 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,d^2=eh,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, 414600);