#507658

order	512
nilpotency class	4
isoclinism	isoclinic class of #501524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

399 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 414854);