#507425

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

548 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,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, 414621);