#507709

order	512
nilpotency class	4
isoclinism	isoclinic class of #501527
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	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

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

normal subgroups only

This group has 471 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)(9,10,16,18,29,36,28,38,13,21,14,19,15,20,12,27)(11,22,23,35,31,33,30,17,26,40,24,39,25,37,32,34);(1,2,4,6)(3,8,7,5)(12,28)(14,16)(18,19)(22,40)(23,24)(27,38)(30,32)(33,37);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,23)(13,26)(14,30)(15,31)(16,32)(18,33)(19,37)(20,39)(21,34)(22,38)(24,28)(25,29)(27,40)(35,36);(9,12,15,14,13,28,29,16)(10,18,36,38,21,19,20,27)(11,23,31,30,26,24,25,32)(17,33,35,22,34,37,39,40)

Polycyclic presentation:

2^9|a^2=fgi,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,d^b=di,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, 414905);