Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18138 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3486 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);(3,4,8,12)(5,13,15,29)(6,17,19,31)(7,10,21,25)(9,22,24,14)(11,26,28,20)(16,27,30,34)(18,23,32,33);(3,5,7,16)(4,9,10,23)(6,18,20,14)(8,15,21,30)(11,27,17,13)(12,24,25,33)(19,32,26,22)(28,34,31,29);(6,19)(11,28)(13,29)(14,22)(17,31)(18,32)(20,26)(27,34);(4,10)(9,23)(11,17)(12,25)(13,27)(24,33)(28,31)(29,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253575);