Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#562 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3820 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)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,25)(19,26,20,24)(21,31,32,22)(27,28,29,30);(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,24,25,26)(21,27,32,29)(22,28,31,30);(6,7)(10,13)(14,15)(17,20,23,19)(18,24,25,26)(21,29,32,27)(22,28,31,30);(1,2,3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,21,23,32)(18,22,25,31)(19,27,20,29)(24,28,26,30);(9,11)(10,14)(12,16)(13,15)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,32)(26,29)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4942431);