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×#50892 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6782 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)(9,10,14,18)(11,19,20,15)(16,22,21,17)(23,24,28,31)(25,29,32,37)(26,27,30,36)(33,38,35,34);(8,13)(9,14)(11,20)(17,22)(25,32)(26,30)(27,36)(29,37);(3,4,5,6)(7,9)(8,10)(11,16)(12,14)(13,18)(15,17)(19,22)(20,21)(23,25)(24,29)(26,35)(27,34)(28,32)(30,33)(31,37)(36,38);(7,10)(8,14)(9,13)(11,17)(12,18)(15,21)(16,19)(20,22)(23,26,28,30)(24,27,31,36)(25,33,32,35)(29,38,37,34);(7,11,12,20)(8,15,13,19)(9,16,14,21)(10,17,18,22)(23,27)(24,30)(25,34)(26,31)(28,36)(29,33)(32,38)(35,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613892);