#3335466

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=fg,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

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, 3242662);