Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#45347 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,7,6)(5,9,10,8)(12,15)(16,17)(19,20,24,27)(21,22,25,30)(23,31,33,26)(28,34,32,29);(11,12)(13,17)(14,15)(16,18)(19,21,24,25)(20,22,27,30)(23,32,33,28)(26,34,31,29);(3,5,7,10)(4,8,6,9)(19,21)(20,22)(23,28)(24,25)(26,29)(27,30)(31,34)(32,33);(3,6,7,4)(5,9,10,8)(11,13,14,18)(12,16,15,17)(19,22)(20,25)(21,27)(23,29)(24,30)(26,32)(28,31)(33,34);(19,23,24,33)(20,26,27,31)(21,28,25,32)(22,29,30,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7577224);