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 | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,8)(9,10,12,11)(13,14)(15,23)(16,22)(17,20)(18,21)(19,25)(24,28)(26,27);(2,4)(13,15,18,26,17,24,16,25)(14,19,22,28,20,27,21,23);(5,6,7,8)(14,20)(19,27)(21,22)(23,28);(9,11,12,10)(15,24)(19,27)(23,28)(25,26)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fi,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72240760, #72240772, #72240776, … |
| maximal normal subgroup of | #72240760, #72240772, #72240776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261751);