Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46175 |
| 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 | |
4590 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);(4,6)(7,10)(12,15)(16,18)(19,20,24,27)(21,28,25,22)(23,26,31,33)(29,34,32,30);(3,4,8,6)(5,10,9,7)(21,25)(22,28)(23,31)(26,33);(3,5)(4,7)(6,10)(8,9)(11,12,14,15)(13,16,17,18)(19,21,24,25)(20,22,27,28)(23,29,31,32)(26,30,33,34);(3,6,8,4)(5,10,9,7)(11,13,14,17)(12,16,15,18)(19,22)(20,25)(21,27)(23,30)(24,28)(26,32)(29,33)(31,34);(3,7,8,10)(4,9,6,5)(19,23)(20,26)(21,29)(22,30)(24,31)(25,32)(27,33)(28,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609938);