Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #94871 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(9,10)(11,17,19,23)(12,15)(13,16)(14,18,21,24)(20,22);(5,6)(7,8)(9,11)(10,14,16,21)(12,18)(13,19)(15,17,22,23)(20,24);(1,2,3,4)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,b^a=bd,c^a=ce,h^a=hi,c^b=cf,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72210893, #72237221, #72237228, … |
| maximal normal subgroup of | #72210893, #72237221, #72237228, … |
| maximal quotient of | #72210873, #72211196, #72238986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31050);