Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36347 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2555 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4,8,6)(5,9,10,7)(19,20,22,24)(21,23,25,26);(3,5)(4,7)(6,9)(8,10)(11,12,14,16)(13,15,18,17)(19,20)(21,23)(22,24)(25,26);(4,6)(5,10)(11,12)(13,17)(14,16)(15,18);(3,6,8,4)(5,9,10,7)(11,13)(12,15)(14,18)(16,17);(3,7,8,9)(4,5,6,10)(11,12)(13,17)(14,16)(15,18)(19,21,22,25)(20,23,24,26)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537197);