#6671006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3013	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=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, 6578202);