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×#44613 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5300 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)(5,7)(6,8)(9,10)(13,17)(16,18)(19,20,23,24)(21,27,28,25)(22,26,30,32)(29,34,31,33);(3,4,6,8)(5,7,9,10)(11,12,15,14)(13,16,17,18)(20,24)(21,28)(22,30)(33,34);(11,13)(12,16)(14,18)(15,17)(19,21,23,28)(20,25,24,27)(22,31,30,29)(26,34,32,33);(3,4)(5,7)(6,8)(9,10)(19,22)(20,26)(21,29)(23,30)(24,32)(25,33)(27,34)(28,31);(3,5,6,9)(4,7,8,10)(11,14,15,12)(13,18,17,16)(19,20)(21,25)(22,26)(23,24)(27,28)(29,33)(30,32)(31,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7576161);