Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37894 |
| 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 | |
5440 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);(5,9)(7,10)(11,12,15,17)(13,14,16,18)(19,20,24,25)(21,23,27,32)(22,29,31,26)(28,34,33,30);(3,4,6,8)(5,10,9,7)(11,13)(12,14)(15,16)(17,18)(20,25)(23,32)(26,29)(30,34);(3,5,6,9)(4,7,8,10)(11,14)(12,16)(13,17)(15,18)(19,21,24,27)(20,23,25,32)(22,28,31,33)(26,30,29,34);(19,22,24,31)(20,26,25,29)(21,28,27,33)(23,30,32,34);(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,23)(20,27)(21,25)(22,30)(24,32)(26,33)(28,29)(31,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542473);