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×#51976 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6956 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);(3,4,5,6)(7,8)(9,13)(10,17)(11,12)(14,18)(15,21)(16,19)(20,22)(23,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,35)(36,38);(8,12)(13,19)(14,17)(20,21)(23,25,28,30)(24,29,32,26)(27,35,36,33)(31,34,38,37);(3,4)(5,6)(7,9,11,16)(8,13,12,19)(10,15,18,22)(14,20,17,21)(23,25)(24,26)(27,33)(28,30)(29,32)(31,34)(35,36)(37,38);(7,8)(9,13)(10,14)(11,12)(15,20)(16,19)(17,18)(21,22)(23,26)(24,30)(25,32)(27,34)(28,29)(31,35)(33,38)(36,37);(7,10)(8,14)(9,15)(11,18)(12,17)(13,20)(16,22)(19,21)(23,27,28,36)(24,31,32,38)(25,33,30,35)(26,34,29,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615210);