Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10)(11,15)(12,16)(13,14)(18,21)(19,23)(22,24);(5,6,8,7)(10,14)(15,16);(9,11,13,12)(10,15,14,16)(17,18,19,21,20,22,23,24);(5,7,8,6)(9,12,13,11)(10,16,14,15)(18,22)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ef,d^2=eh,f^2=i,b^a=be,c^a=cf,f^a=fi,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72240744, #72240760, #72240761, … |
| maximal normal subgroup of | #72240744, #72240760, #72240761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262207);