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×#25198 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3031 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);(3,4,7,12,8,13,23,11)(5,14,17,25,18,27,16,9)(6,19,21,26,22,28,20,10)(15,24,30,33,31,34,32,29);(3,5,7,17,8,18,23,16)(4,9,12,14,13,25,11,27)(6,15,21,30,22,31,20,32)(10,24,19,33,26,34,28,29);(3,6,8,22)(4,10,13,26)(5,15,18,31)(7,20,23,21)(9,24,25,34)(11,19,12,28)(14,29,27,33)(16,30,17,32);(3,4)(5,9)(6,10)(7,12)(8,13)(11,23)(14,17)(15,24)(16,27)(18,25)(19,21)(20,28)(22,26)(29,32)(30,33)(31,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346097);