Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18131 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×#120 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,11)(5,12,14,17)(6,10,7,15)(9,16,18,13)(24,30)(31,34);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15)(19,20)(21,24)(22,25)(23,26)(27,31)(28,30)(29,32)(33,34);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(19,21)(20,24)(22,27)(23,28)(25,31)(26,30)(29,33)(32,34);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(19,22,23,29)(20,25,26,32)(21,27,28,33)(24,31,30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253566);