#512276

order	512
nilpotency class	4
isoclinism	isoclinic class of #511896
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	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^c=gi,f^d=fi,g^d=ghi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419472);