Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#61097 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4)(5,6)(7,8)(9,17)(10,14)(11,24)(12,15)(13,20)(16,27)(18,35)(19,33)(21,38)(22,34)(23,31)(25,29)(26,32)(28,37)(30,36);(7,9,12,19)(8,13,15,29)(10,21,23,37)(11,25,27,20)(14,30,31,35)(16,33,24,17)(18,26,36,22)(28,34,38,32);(4,6)(7,8)(9,13)(10,14)(11,16)(12,15)(17,20)(18,28)(19,29)(21,30)(22,32)(23,31)(24,27)(25,33)(26,34)(35,37)(36,38);(7,10,12,23)(8,14,15,31)(9,18,19,36)(11,26,27,22)(13,28,29,38)(16,34,24,32)(17,35,33,30)(20,37,25,21);(8,15)(9,19)(14,31)(16,24)(18,36)(20,25)(21,37)(32,34)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483285);