Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15310 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,7)(10,11)(12,16)(18,21)(19,20)(25,26)(27,28)(29,30)(31,32);(1,2)(3,7)(4,5)(6,8)(10,11)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32);(9,10,13,11)(12,14,16,15)(17,19,22,20)(18,23,21,24)(25,28)(26,32)(27,29)(30,31);(1,2)(3,7)(4,5)(6,8)(9,11,13,10)(12,15,16,14)(17,20,22,19)(18,23,21,24)(25,28)(26,27)(29,32)(30,31);(1,3,4,8)(2,6,5,7)(9,12)(10,14)(11,15)(13,16)(17,21,22,18)(19,24,20,23)(25,29,30,26)(27,31,32,28)Polycyclic presentation:
2^9|c^2=fghi,d^2=fghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2857477);