Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12842 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,12,16,13,10,24,17)(9,18,22,33,23,35,38,34)(11,26,19,15,27,31,36,25)(14,28,29,32,20,37,30,21);(7,9,19,28,12,22,27,32,13,23,36,37,24,38,11,21)(8,14,26,34,17,30,25,35,10,20,31,33,16,29,15,18);(4,6)(7,10)(8,13)(9,20)(11,25)(12,16)(14,23)(15,27)(17,24)(18,32)(19,31)(21,35)(22,29)(26,36)(28,33)(30,38)(34,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401021);