#507605

order	512
nilpotency class	4
isoclinism	isoclinic class of #500599
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 ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

417 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=g,c^2=gh,d^2=ei,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414801);