Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#23603 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8,13,16,12,17,11,25)(9,18,27,29,20,35,21,14)(10,15,26,31,24,32,23,33)(19,34,37,28,36,30,22,38);(3,4)(5,6)(7,9,11,21,12,20,13,27)(8,14,16,18,17,29,25,35)(10,22,23,36,24,37,26,19)(15,30,31,38,32,34,33,28);(4,6)(7,10,12,24)(8,15,17,32)(9,19,20,36)(11,26,13,23)(14,28,29,38)(16,33,25,31)(18,34,35,30)(21,37,27,22);(9,20)(14,29)(18,35)(19,36)(21,27)(22,37)(28,38)(30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339313);