Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,10,9)(7,12,8,11)(15,19)(18,20)(21,22,24,26)(23,27,28,25);(2,4)(5,7,10,8)(6,11,9,12)(14,17)(18,20)(21,22,24,26)(23,25,28,27);(1,2,3,4)(7,8)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28);(5,8,10,7)(6,11,9,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,25)(24,28)(26,27);(5,9,10,6)(7,11,8,12)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fhi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 612418);