Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,16,19)(15,17,20,18)(21,22,27,24)(23,25,31,28)(26,29,34,32)(30,33,36,35);(7,11)(10,12)(13,15)(14,17)(16,20)(18,19)(21,23,27,31)(22,28,24,25)(26,30,34,36)(29,35,32,33);(6,9)(10,12)(13,15,16,20)(14,18,19,17)(21,24,27,22)(23,28,31,25)(26,29,34,32)(30,33,36,35);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,25,27,28)(22,23,24,31)(26,33,34,35)(29,30,32,36);(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,19)(17,20)(21,26,27,34)(22,29,24,32)(23,30,31,36)(25,33,28,35)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5129462);