Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#61100 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C22 | C2×D42 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,15)(6,17)(7,11)(8,12)(9,13)(10,14)(16,31)(18,20)(19,32)(21,25)(22,34)(23,28)(24,30)(26,29)(27,33);(3,5)(4,6)(7,22)(8,17)(9,25)(10,18)(11,27)(12,20)(13,15)(14,21)(16,23)(19,29)(24,32)(26,33)(28,34)(30,31);(3,6)(4,5)(7,19)(8,20)(9,15)(10,21)(11,16)(12,17)(13,25)(14,18)(22,29)(23,27)(24,34)(26,31)(28,32)(30,33);(3,7)(4,11)(5,16)(6,19)(8,23)(9,26)(10,24)(12,28)(13,29)(14,30)(15,31)(17,32)(18,33)(20,27)(21,34)(22,25);(3,8,10,9)(4,12,14,13)(5,17,18,25)(6,20,21,15)(7,23,24,26)(11,28,30,29)(16,32,33,22)(19,27,34,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483360);