Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C2×#24378 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7264 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)(11,12,13,16,14,17,18,15);(4,6)(7,8)(9,10);(8,10)(12,15)(13,18)(16,17);(4,6)116 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6437748, C2×#6573561, C2×D4×#447, … |
| maximal normal subgroup of | C2×#6437748, C2×#6573561, C2×D4×#447, … |
| maximal quotient of | #72305928, #72306033, #72306089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344921);