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×#52066 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2733 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6934 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)(9,13,16,11)(10,14,18,22)(17,21,19,20)(23,24)(25,28)(26,34)(27,30)(29,35)(31,33)(32,37)(36,38);(9,16)(10,18)(11,13)(14,22)(23,24,27,30)(25,28,33,31)(26,34,35,29)(32,37,38,36);(3,4,5,6)(7,9)(8,13)(10,19)(11,15)(12,16)(14,20)(17,18)(21,22)(23,24)(25,31)(26,29)(27,30)(28,33)(32,37)(34,35)(36,38);(7,10,12,18)(8,14,15,22)(9,17,16,19)(11,20,13,21)(23,25)(24,28)(26,32)(27,33)(29,36)(30,31)(34,37)(35,38);(7,11,12,13)(8,9,15,16)(10,20,18,21)(14,17,22,19)(23,26,27,35)(24,29,30,34)(25,32,33,38)(28,36,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615334);