Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C24 | C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14)(15,20)(16,19)(17,21)(18,22)(23,28)(24,26)(25,27);(2,4)(14,19,22,21)(15,23)(20,27,28,26)(24,25);(6,8)(13,15,17,25,18,23,16,24)(14,20,21,27,22,28,19,26)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72213009, #72215990, #72215994, … |
| maximal normal subgroup of | #72213009, #72215990, #72215994, … |
| maximal quotient of | #72212999 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30633);