Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25145 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3150 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);(3,4)(5,6)(7,8,13,26,12,17,11,16)(9,14,27,38,20,30,19,29)(10,21,23,15,25,32,24,33)(18,34,35,28,22,37,36,31);(4,6)(7,9,11,19,12,20,13,27)(8,14,16,29,17,30,26,38)(10,22,24,35,25,18,23,36)(15,31,21,37,33,28,32,34);(7,10)(8,15)(9,18)(11,23)(12,25)(13,24)(14,28)(16,32)(17,33)(19,35)(20,22)(21,26)(27,36)(29,37)(30,31)(34,38);(4,6)(7,11,12,13)(8,16,17,26)(9,19,20,27)(10,23,25,24)(14,29,30,38)(15,32,33,21)(18,35,22,36)(28,37,31,34)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346071);