Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32870 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,11,9,14,8,13)(5,15,18,10,19,25,16,26)(6,12,22,27,23,28,21,29)(17,30,32,24,20,33,31,34);(3,5,7,18,9,19,8,16)(4,10,13,15,14,26,11,25)(6,20,21,32,23,17,22,31)(12,24,27,33,28,34,29,30);(4,11)(5,16)(6,21)(7,8)(12,28)(13,14)(15,25)(17,20)(18,19)(22,23)(24,33)(30,34);(3,6)(4,12)(5,17)(7,22)(8,21)(9,23)(10,24)(11,27)(13,29)(14,28)(15,30)(16,31)(18,32)(19,20)(25,33)(26,34)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538648);