#6440372

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437473
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	#62811	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4238 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)(11,22)(12,18)(13,19)(14,31)(15,21)(16,20)(17,25)(23,39)(24,32)(26,34)(27,38)(28,36)(29,35)(30,33)(37,40);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,24)(13,30)(14,26)(15,25)(16,27)(18,33)(19,37)(20,34)(21,38)(22,31)(23,28)(29,40)(32,36)(35,39);(2,6)(5,8)(9,12,16,29)(10,18,20,35)(11,23,27,30)(13,14,28,15)(17,32,34,37)(19,31,36,21)(22,39,38,33)(24,26,40,25);(9,13)(10,19)(11,24)(12,14)(15,29)(16,28)(17,33)(18,31)(20,36)(21,35)(22,32)(23,26)(25,30)(27,40)(34,39)(37,38);(10,20)(17,34)(18,35)(19,36)(21,31)(22,38)(32,37)(33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312791, #72313120, #72313127
maximal normal subgroup of	#72312791, #72313120, #72313127
maximal quotient of	#72316992, #72317035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347568);