#507426

order	512
nilpotency class	4
isoclinism	isoclinic class of #501764
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414622);