Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12925 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| 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 | |
1078 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);(3,4,6,5)(7,8)(9,20)(10,24)(11,18)(12,17)(13,19)(14,21)(15,22)(16,27)(23,35)(25,36)(26,30)(28,34)(29,33)(31,38)(32,37);(3,5,6,4)(7,9)(8,14)(10,25)(11,22)(12,29)(13,23)(15,30)(16,33)(17,31)(18,36)(19,32)(20,24)(21,27)(26,37)(28,38)(34,35);(7,9)(8,15)(10,21)(11,29)(12,22)(13,23)(14,30)(16,32)(17,35)(18,20)(19,33)(24,36)(25,27)(26,38)(28,37)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401449);