#512358

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
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	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

495 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=efi,d^2=eh,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,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,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, 419554);