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×#48819 |
| 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 | |
6858 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)(7,8,12,15)(9,10,13,17)(11,14,19,21)(16,18,20,22)(23,24,28,32)(25,26,30,29)(27,35,37,31)(33,38,36,34);(9,13)(10,17)(11,19)(14,21)(23,25,28,30)(24,29,32,26)(27,36,37,33)(31,34,35,38);(3,4,5,6)(7,9)(8,10)(11,16)(12,13)(14,18)(15,17)(19,20)(21,22)(23,25)(24,29)(26,32)(27,33)(28,30)(31,38)(34,35)(36,37);(7,10)(8,13)(9,15)(11,18)(12,17)(14,20)(16,21)(19,22)(23,26,28,29)(24,30,32,25)(27,34,37,38)(31,36,35,33);(7,11)(8,14)(9,16)(10,18)(12,19)(13,20)(15,21)(17,22)(23,27)(24,31)(25,33)(26,34)(28,37)(29,38)(30,36)(32,35)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7611672);