#502760

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61284	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

831 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 831 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,24)(13,26)(14,17)(15,19)(16,22)(18,29)(21,36)(23,32)(25,34)(27,39)(28,31)(30,33)(35,38)(37,40);(2,5)(6,7)(11,21)(12,14)(13,27)(16,31)(17,24)(18,34)(20,36)(22,28)(23,38)(25,29)(26,39)(30,37)(32,35)(33,40);(9,11,12,22,15,23,14,30)(10,16,17,20,19,33,24,32)(13,28,25,21,29,37,27,38)(18,35,34,40,26,36,39,31);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|c^2=f,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,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, 409956);