Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#24986 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | #2420 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1828 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)(5,16)(6,20)(7,11)(8,25)(9,14)(10,15)(12,23)(13,19)(17,26)(18,27)(21,28)(22,30)(24,29)(31,32)(33,34);(3,5,8,7,10,19,9,18)(4,11,14,16,15,27,25,13)(6,21,23,17,24,32,22,33)(12,28,20,34,30,31,29,26);(3,6)(4,12)(5,17)(7,21)(8,22)(9,23)(10,24)(11,26)(13,28)(14,29)(15,30)(16,31)(18,32)(19,33)(20,25)(27,34);(3,7,8,19,10,18,9,5)(4,13,14,11,15,16,25,27)(6,21,22,33,24,32,23,17)(12,28,29,26,30,31,20,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354251);