Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#60432 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10296 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);(4,6)(7,8,12,16)(9,17,20,30)(10,14,21,28)(11,23,25,15)(13,24,27,19)(18,31,32,38)(22,35,34,29)(26,36,37,33);(3,4,5,6);(7,9,12,20)(8,13,16,27)(10,18,21,32)(11,24,25,19)(14,26,28,37)(15,30,23,17)(22,36,34,33)(29,38,35,31);(7,10)(8,14)(9,18)(11,22)(12,21)(13,26)(15,29)(16,28)(17,31)(19,33)(20,32)(23,35)(24,36)(25,34)(27,37)(30,38);(10,21)(14,28)(18,32)(22,34)(26,37)(29,35)(31,38)(33,36)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482150);