Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#32947 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4494 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);(4,9)(7,18)(10,19)(11,20)(12,21)(13,25)(14,27)(16,28)(22,32)(23,30)(24,31)(29,33);(3,4,7,9,8,12,18,21)(5,10,14,19,15,23,27,30)(6,11,16,20,17,24,28,31)(13,22,26,29,25,33,34,32);(3,5)(4,10)(6,13)(7,14)(8,15)(9,19)(11,22)(12,23)(16,26)(17,25)(18,27)(20,29)(21,30)(24,33)(28,34)(31,32);(3,6)(4,11)(5,13)(7,16)(8,17)(9,20)(10,22)(12,24)(14,26)(15,25)(18,28)(19,29)(21,31)(23,33)(27,34)(30,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536237);