Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(10,13)(11,15)(17,18,22,25)(19,26,24,21)(20,28,30,23)(27,31,32,29);(6,8)(11,15)(14,16)(17,19,22,24)(18,21,25,26)(20,27,30,32)(23,31,28,29);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,27)(22,24)(23,31)(25,26)(28,29)(30,32);(1,2,3,4)(9,11,12,15)(10,14,13,16)(17,20,22,30)(18,23,25,28)(19,27,24,32)(21,31,26,29);(9,10)(11,14)(12,13)(15,16)(17,21)(18,24)(19,25)(20,29)(22,26)(23,27)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3370988);