Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60875 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3494 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(20,25)(21,29)(22,24)(26,33)(30,34)(31,32);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,20,23,25)(21,30,31,26)(22,27,24,28)(29,33,32,34);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,21,23,31)(20,26,25,30)(22,29,24,32)(27,34,28,33);(4,10)(8,11)(9,18)(16,17)(19,22,23,24)(20,27,25,28)(21,29,31,32)(26,34,30,33);(20,25)(26,30)(27,28)(33,34)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482913);