Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#34639 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| socle | C24 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12)(13,15)(14,16)(17,18)(19,20)(21,24)(22,26)(23,25)(27,30)(28,31)(29,32)(33,34);(3,5,7,6)(4,8,9,10)(11,13,14,17)(12,15,16,18)(19,21)(20,24)(22,28)(23,27)(25,30)(26,31)(29,33)(32,34);(3,6,7,5)(4,8,9,10)(12,16)(15,18)(21,27)(24,30)(28,33)(31,34);(20,25)(24,30)(26,32)(31,34);(19,22,23,29)(20,26,25,32)(21,28,27,33)(24,31,30,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534995);