Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13084 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,10,14,11,13,19,20)(9,15,17,22,16,12,18,21);(7,9,10,18)(8,12,20,22)(11,16,19,17)(13,15,14,21);(4,6)74 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#499797, #72233501, #72233512, … |
| maximal normal subgroup of | C2×#499797, #72233501, #72233512, … |
| maximal quotient of | #72233479, #72241064, #72241091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401886);