Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25197 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3063 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);(3,4,8,12,9,13,7,21)(5,14,16,10,17,25,22,26)(6,18,19,28,20,27,23,11)(15,30,32,34,33,24,31,29);(3,5,8,16,9,17,7,22)(4,10,12,25,13,26,21,14)(6,15,19,32,20,33,23,31)(11,24,18,29,28,30,27,34);(3,6)(4,11)(5,15)(7,19)(8,23)(9,20)(10,24)(12,27)(13,28)(14,29)(16,31)(17,33)(18,21)(22,32)(25,34)(26,30);(3,4,9,13)(5,10,17,26)(6,11,20,28)(7,21,8,12)(14,16,25,22)(15,24,33,30)(18,23,27,19)(29,31,34,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346095);