Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#13090 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2928 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)(9,15,17,22)(10,13,19,21)(11,14)(12,16)(18,20);(7,9,10,18)(8,12,13,15)(11,17,19,16)(14,20,21,22);(4,6)74 conjugacy classes — view the full character table.
| maximal subgroup of | #72249387, #72249389, C2×#500059, … |
| maximal normal subgroup of | #72249387, #72249389, C2×#500059, … |
| maximal quotient of | #72263161, #72282346, #72284058, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401905);