Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36059 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5676 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);(4,7)(5,9)(11,12,15,13)(14,17,18,16)(19,20,24,22)(21,25,27,28)(23,26,31,30)(29,32,33,34);(3,4)(5,8)(6,7)(9,10)(11,13,15,12)(14,17,18,16)(21,27)(25,28)(29,33)(32,34);(3,5,6,9)(4,8,7,10)(11,14)(12,16)(13,17)(15,18)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,34)(31,33);(19,22,24,20)(21,28,27,25)(23,26,31,30)(29,32,33,34);(19,23,24,31)(20,26,22,30)(21,29,27,33)(25,32,28,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536861);