#507704

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 478 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)(9,10,15,18,29,38,30,35,14,20,13,28,16,19,33,17)(11,12,21,25,39,34,40,24,22,27,31,36,23,26,37,32);(1,2,4,6)(3,8,7,5)(9,11)(10,12)(13,21)(14,22)(15,31)(16,23)(17,24)(18,36)(19,26)(20,27)(25,28)(29,39)(30,37)(32,35)(33,40)(34,38);(1,3,4,7)(2,5,6,8)(9,12)(10,11)(13,24)(14,27)(15,32)(16,34)(17,21)(18,37)(19,39)(20,22)(23,38)(25,33)(26,29)(28,40)(30,36)(31,35);(9,13,29,33,14,15,16,30)(10,17,19,28,20,35,38,18)(11,21,39,40,22,31,23,37)(12,24,26,25,27,32,34,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^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, 414900);