#508422

order	512
nilpotency class	4
isoclinism	isoclinic class of #508286
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	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,d^c=de,f^c=fi,g^c=gh,h^c=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, 415618);