#6470040

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377236);