#6579438

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

Polycyclic presentation:

2^9|b^2=gi,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=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, 6486634);