#512274

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
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×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 502 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

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,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, 419470);