Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C227 |
| direct product | C2×#63103 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C26 |
| commutator, Frattini, cosocle | C2 | C28 |
| socle | C22 | C27 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
420694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 8 generators on 34 points:
(1,2);(3,4)(5,13)(6,10)(7,12)(8,11)(9,15)(14,25)(16,26)(17,22)(18,23)(19,24)(20,27)(21,29)(28,33)(30,32)(31,34);(3,5)(4,9)(6,14)(7,16)(8,15)(10,20)(11,13)(12,21)(17,27)(18,28)(19,29)(22,25)(23,31)(24,26)(30,34)(32,33);(3,6)(4,10)(5,14)(7,18)(8,17)(9,20)(11,22)(12,23)(13,25)(15,27)(16,28)(19,30)(21,31)(24,32)(26,33)(29,34);(6,17)(10,22)(14,27)(18,30)(20,25)(23,32)(28,34)(31,33);(5,15)(9,13)(14,27)(16,29)(20,25)(21,26)(28,34)(31,33);(4,11)(9,13)(10,22)(12,24)(20,25)(21,26)(23,32)(31,33);(3,7,8,19)(4,12,11,24)(5,16,15,29)(6,18,17,30)(9,21,13,26)(10,23,22,32)(14,28,27,34)(20,31,25,33)260 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10494210);