Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8608 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C23×D4 | Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1346 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)(9,10,13,17,14,18,12,16)(11,19,21,24,22,23,20,15);(9,11,12,20,14,22,13,21)(10,15,17,19,18,24,16,23);(1,3)(2,5)(4,8)(6,7);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280403, #72280443, #72280475, … |
| maximal normal subgroup of | #72280403, #72280443, #72280475, … |
| maximal quotient of | C2×#188834 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96030);