Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,12,10,7,8,11)(13,14)(15,18,22,24)(16,20)(17,19)(21,25,27,28)(23,26);(5,7,9,11,10,6,8,12)(13,15)(14,18,19,24)(16,21)(17,22)(20,25,26,28)(23,27);(1,2,3,4)(14,19)(18,24)(20,26)(25,28);(5,8,10,9)(6,11,7,12)(13,16,17,23)(14,20,19,26)(15,21,22,27)(18,25,24,28)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=fh,d^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111363);