Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18742 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,14)(12,17)(13,15)(16,18)(20,25)(21,28)(22,23)(27,31)(29,30)(32,33);(4,8)(6,14)(9,15)(10,11)(13,18)(16,17)(19,20)(21,26)(22,25)(23,31)(24,27)(28,32)(29,33)(30,34);(3,5,7,12)(4,9,10,17)(6,13,14,18)(8,15,11,16)(19,21,24,30)(20,26,27,34)(22,28,23,29)(25,32,31,33);(4,10)(5,12)(8,11)(13,18)(19,22,24,23)(20,25,27,31)(21,29,30,28)(26,33,34,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255549);