Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | |
2012 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)(5,6,8,10)(7,11,12,9)(15,19)(18,20)(21,22,27,31)(23,29,24,28)(25,26,36,30)(32,35,34,33);(14,17)(18,20)(21,23,27,24)(22,28,31,29)(25,32,36,34)(26,33,30,35);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24,27,23)(22,29,31,28)(25,34,36,32)(26,35,30,33);(1,2,3,4)(13,15)(14,18)(16,19)(17,20)(21,25,27,36)(22,30,31,26)(23,32,24,34)(28,35,29,33);(5,7,8,12)(6,9,10,11)(21,26,27,30)(22,25,31,36)(23,33,24,35)(28,32,29,34)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1202243);