Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51389 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5402 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);(4,6)(8,9)(11,12)(13,17)(14,16)(15,18)(20,24)(25,28)(26,31)(32,34);(3,4,7,6)(5,9,10,8)(11,12,14,16)(13,15,18,17)(20,24)(21,27)(22,30)(32,34);(11,13)(12,15)(14,18)(16,17)(19,20,23,24)(21,28,27,25)(22,26,30,31)(29,34,33,32);(3,5,7,10)(4,8,6,9)(11,12)(13,15)(14,16)(17,18)(19,21)(20,25)(22,29)(23,27)(24,28)(26,32)(30,33)(31,34);(3,6,7,4)(5,9,10,8)(11,13)(12,15)(14,18)(16,17)(19,22,23,30)(20,26,24,31)(21,29,27,33)(25,32,28,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613961);