Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#50891 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6896 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)(7,8,12,13)(9,15,17,10)(11,19,20,14)(16,18,22,21)(23,24)(25,31)(26,30)(27,28)(29,32)(33,38)(34,37)(35,36);(8,13)(10,15)(14,19)(18,21)(24,27)(25,32)(30,35)(33,34);(3,4,5,6)(7,9)(8,10)(11,16)(12,17)(13,15)(14,18)(19,21)(20,22)(23,25)(24,29)(26,34)(27,31)(28,32)(30,38)(33,36)(35,37);(7,10)(8,9)(11,18)(12,15)(13,17)(14,16)(19,22)(20,21)(23,26,28,36)(24,30,27,35)(25,33,32,34)(29,37,31,38);(7,11,12,20)(8,14,13,19)(9,16,17,22)(10,18,15,21)(23,27,28,24)(25,31,32,29)(26,35,36,30)(33,38,34,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613876);