Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2510 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)(9,10,13,14)(11,15,16,12)(17,18,22,25)(19,21,27,23)(20,28,31,24)(26,32,29,30);(6,8)(10,14)(12,15)(17,19,22,27)(18,23,25,21)(20,29,31,26)(24,30,28,32);(5,6)(7,8)(9,11,13,16)(10,15,14,12)(17,19)(18,23)(20,29)(21,25)(22,27)(24,30)(26,31)(28,32);(1,2,3,4)(9,11)(10,12)(13,16)(14,15)(17,20,22,31)(18,24,25,28)(19,26,27,29)(21,30,23,32);(9,12)(10,11)(13,15)(14,16)(17,21,22,23)(18,19,25,27)(20,30,31,32)(24,26,28,29)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fg,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1455279);