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×#50254 |
| 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 | |
5354 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)(5,10)(6,9)(7,8)(11,12)(13,17)(14,18)(15,16)(19,20,24,27)(21,23,26,32)(22,25,29,33)(28,31,30,34);(3,5,8,6)(4,9,7,10)(11,13)(12,14)(15,18)(16,17)(21,26)(22,29)(23,32)(25,33);(5,6)(9,10)(12,16)(13,18)(19,21,24,26)(20,23,27,32)(22,30,29,28)(25,34,33,31);(3,6,8,5)(4,9,7,10)(11,13)(12,17)(14,16)(15,18)(19,22)(20,25)(21,28)(23,31)(24,29)(26,30)(27,33)(32,34);(3,7,8,4)(5,9,6,10)(11,14,15,17)(12,13,16,18)(19,23)(20,26)(21,27)(22,31)(24,32)(25,30)(28,33)(29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613395);