Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13092 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2164 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)(7,8,11,14,12,15,22,18)(9,16,13,21,19,10,20,17);(7,9,17,8,12,19,21,15)(10,18,22,20,16,14,11,13);(4,6)74 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#499867, #72266735, #72266739, … |
| maximal normal subgroup of | C2×#499867, #72266735, #72266739, … |
| maximal quotient of | #72266739, #72266748, #72266749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401922);