#6579441

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574086
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61088	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,18)(13,15)(14,20)(16,26)(17,31)(19,22)(23,37)(24,38)(25,28)(27,34)(29,39)(30,32)(33,35)(36,40);(2,5)(6,8)(9,11,22,27,15,28,14,26)(10,16,20,25,13,34,19,21)(12,29,32,23,33,24,31,40)(17,38,35,37,30,39,18,36);(1,3,4,7)(2,6,5,8)(9,12,14,31,15,33,22,32)(10,17,19,18,13,30,20,35)(11,23,26,29,28,40,27,24)(16,36,21,39,34,37,25,38);(9,12,15,33)(10,18,13,35)(11,24,28,29)(14,31,22,32)(16,37,34,36)(17,19,30,20)(21,38,25,39)(23,27,40,26);(9,13)(10,15)(11,25)(12,30)(14,20)(16,26)(17,33)(18,32)(19,22)(21,28)(23,38)(24,37)(27,34)(29,36)(31,35)(39,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6486637);