#504341

order	512
nilpotency class	4
isoclinism	isoclinic class of #504125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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 ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=efh,d^2=ei,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,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, 411537);