Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×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 | |
1758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,8)(9,10,12,11)(13,14,18,23)(15,25,26,21)(16,20,27,28)(17,24,19,22);(2,4)(13,15,19,27,18,26,17,16)(14,20,24,21,23,28,22,25);(5,6,7,8)(13,16,17,26,18,27,19,15)(14,21,22,20,23,25,24,28);(9,11,12,10)(15,26)(16,27)(20,28)(21,25)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=gi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72264864, #72264868, #72264943, … |
| maximal normal subgroup of | #72264864, #72264868, #72264943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261792);