#512186

order	512
nilpotency class	4
isoclinism	isoclinic class of #511890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33582	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

624 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 150 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)(10,17)(11,23)(12,27)(13,14)(15,26)(19,35)(20,22)(21,32)(25,30)(28,31)(29,39)(33,34)(36,40)(37,38);(2,5)(6,8)(9,10,13,17,16,22,14,20)(11,24,15,21,26,18,23,32)(12,19,28,33,30,37,29,36)(25,40,31,38,27,34,39,35);(1,3,4,7)(2,6,5,8)(9,11,13,15,16,26,14,23)(10,18,17,32,22,24,20,21)(12,25,28,31,30,27,29,39)(19,34,33,35,37,40,36,38);(9,12)(10,19)(11,25)(13,28)(14,29)(15,31)(16,30)(17,33)(18,34)(20,36)(21,38)(22,37)(23,39)(24,40)(26,27)(32,35)

Polycyclic presentation:

2^9|b^2=fi,c^2=e,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gh,h^a=hi,c^b=ci,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, 419382);