#6470074

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
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

4066 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)(9,10,16,21)(11,22,17,31)(12,24,27,25)(13,18,30,35)(14,19,15,20)(23,36,32,39)(26,38,40,37)(28,33,29,34);(1,3)(2,5)(4,7)(6,8)(9,11,16,17)(10,12,21,27)(13,23,30,32)(14,24,15,25)(18,26,35,40)(19,22,20,31)(28,37,29,38)(33,39,34,36);(3,7)(5,8)(9,12)(10,17)(11,21)(13,26)(14,22)(15,31)(16,27)(18,32)(19,25)(20,24)(23,35)(28,39)(29,36)(30,40)(33,38)(34,37);(3,7)(5,8)(9,13)(10,18)(11,23)(12,26)(14,29)(15,28)(16,30)(17,32)(19,34)(20,33)(21,35)(22,36)(24,38)(25,37)(27,40)(31,39);(2,6)(5,8)(9,14,16,15)(10,19,21,20)(11,24,17,25)(12,22,27,31)(13,28,30,29)(18,33,35,34)(23,37,32,38)(26,39,40,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,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, 6377270);