Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25447 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2688 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);(3,4,8,12,9,13,7,24)(5,14,18,15,19,28,17,10)(6,20,22,26,23,33,21,11)(16,29,31,34,32,25,30,27);(3,5)(4,10,13,15)(6,16)(7,17)(8,18)(9,19)(11,25,26,29)(12,14,24,28)(20,27,33,34)(21,30)(22,31)(23,32);(3,4)(5,15)(6,11)(7,12)(8,24)(9,13)(10,19)(14,18)(16,29)(17,28)(20,21)(22,33)(23,26)(25,32)(27,31)(30,34);(3,6,8,22,9,23,7,21)(4,11,24,33,13,26,12,20)(5,16,18,31,19,32,17,30)(10,25,28,34,15,29,14,27)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351589);