#6579420

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574084
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	#61089	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1938 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)(11,22)(12,29)(13,18)(14,20)(15,21)(16,26)(17,30)(19,33)(23,38)(24,39)(25,28)(27,36)(31,34)(32,37)(35,40);(2,5)(6,8)(9,11,14,26,15,28,33,27)(10,16,19,22,21,36,20,25)(12,24,30,40,13,32,31,23)(17,37,29,35,34,39,18,38);(1,3,4,7)(2,6,5,8)(9,12,14,30,15,13,33,31)(10,17,19,29,21,34,20,18)(11,23,26,24,28,40,27,32)(16,35,22,39,36,38,25,37);(9,13)(10,18)(11,24)(12,15)(14,31)(16,35)(17,19)(20,34)(21,29)(22,39)(23,27)(25,37)(26,40)(28,32)(30,33)(36,38);(9,10)(11,25)(12,17)(13,34)(14,19)(15,21)(16,26)(18,31)(20,33)(22,28)(23,37)(24,35)(27,36)(29,30)(32,38)(39,40)

Polycyclic presentation:

2^9|b^2=g,c^2=f,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=eh,h^b=hi,d^c=dh,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, 6486616);