Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×C22⋊C8 |
| prime radical | C23×D4 | C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,13,18,21,24,14,16,11,15,19,22,20,23);(1,3)(2,5)(4,8)(6,7);(9,11,13,20,14,12,19,21)(10,15,18,23,16,17,22,24);(10,16)(15,17)(18,22)(23,24)Polycyclic presentation:
2^9|a^2=f,c^2=h,f^2=h,h^2=i,b^a=be,d^a=di,e^a=eg,f^b=fg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72280424, #72280443, #72280466, … |
| maximal normal subgroup of | #72280424, #72280443, #72280466, … |
| maximal quotient of | C2×#480075 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387271);