Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13004 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,11,15,21,19,10,20,13,16,22,18,12,9,17,14);(7,9,12,8,13,19,21,16)(10,20,11,18,17,14,22,15);(4,6)92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250015, #72250034, #72250042, … |
| maximal normal subgroup of | #72250015, #72250034, #72250042, … |
| maximal quotient of | #72265420, #72280693, C2×#136613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401667);