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×#39250 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4786 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);(13,17)(16,18)(19,20)(21,28)(22,23)(24,25)(26,31)(27,32)(29,33)(30,34);(3,4,6,8)(5,9,10,7)(11,12,15,14)(13,16,17,18)(20,25)(23,27)(26,28)(30,33);(3,4)(5,7)(6,8)(9,10)(11,13)(12,16)(14,18)(15,17)(19,21,24,31)(20,26,25,28)(22,29,32,34)(23,30,27,33);(11,14,15,12)(13,18,17,16)(19,22)(20,27)(21,29)(23,25)(24,32)(26,33)(28,30)(31,34);(3,5,6,10)(4,7,8,9)(19,23,24,27)(20,22,25,32)(21,30,31,33)(26,29,28,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544259);