#508019

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
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	C₂×#3061	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1010 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 148 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)(9,10,12,17)(11,20,22,39)(13,27,26,38)(14,19,15,31)(16,32,33,23)(18,29,34,30)(21,28,36,37)(24,35,25,40);(2,5)(6,8)(9,10)(11,21)(12,17)(13,18)(14,31)(15,19)(16,24)(20,37)(22,36)(23,35)(25,33)(26,34)(27,30)(28,39)(29,38)(32,40);(1,3)(2,6)(4,7)(5,8)(9,11,12,22)(10,16,17,33)(13,28,26,37)(14,25,15,24)(18,35,34,40)(19,36,31,21)(20,38,39,27)(23,29,32,30);(3,7)(6,8)(11,22)(16,33)(20,39)(21,36)(23,32)(24,25)(28,37)(35,40)

Polycyclic presentation:

2^9|a^2=i,c^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415215);