Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C2×#23919 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#121 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
8438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,6)(11,12,14,17)(13,15,18,16);(4,6)(7,8)(9,10);(8,10)(12,15)(13,18)(16,17);(4,6)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×#450, C2×#6573561, C2×#6437412, … |
| maximal normal subgroup of | C2×D4×#450, C2×#6573561, C2×#6437412, … |
| maximal quotient of | #72261247, #72305817, #72306038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344595);