Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C4×#95 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,16)(12,15)(13,17)(14,18)(19,24)(20,22)(21,23);(1,3)(2,5)(4,8)(6,7)(10,15,18,17)(11,19)(16,23,24,22)(20,21);(3,8)(5,7)(9,11,13,21,14,19,12,20)(10,16,17,23,18,24,15,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=hi,g^2=hi,b^a=bd,d^a=dh,f^a=fhi,c^b=ch,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280575, C2×#105326 |
| maximal normal subgroup of | #72280575, C2×#105326 |
| maximal quotient of | C2×#105326 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12522);