Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#41414 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2812 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(4,6)(7,8,12,13)(9,10,15,18)(11,14,19,20)(16,17,21,22)(23,24)(25,32)(26,27)(28,29)(30,35)(31,37)(33,34)(36,38);(8,13)(9,15)(14,20)(16,21)(24,29)(27,31)(30,32)(34,38);(3,4,5,6)(7,9)(8,10)(11,16)(12,15)(13,18)(14,17)(19,21)(20,22)(23,25,28,35)(24,30,29,32)(26,36,37,33)(27,34,31,38);(7,10)(8,9)(11,17)(12,18)(13,15)(14,16)(19,22)(20,21)(23,26)(24,31)(25,33)(27,29)(28,37)(30,34)(32,38)(35,36);(7,11,12,19)(8,14,13,20)(9,16,15,21)(10,17,18,22)(23,27,28,31)(24,26,29,37)(25,34,35,38)(30,36,32,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547084);