Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C8×D4 |
| prime radical | C24 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(5,6,7,8)(9,10,14,19,16,20,12,29,17,22,15,21,13,32,31,37)(11,18,25,35,27,34,30,40,28,36,26,33,24,39,23,38);(2,4)(9,11,15,26,16,27,31,23,17,28,14,25,13,24,12,30)(10,18,21,33,20,34,37,38,22,36,19,35,32,39,29,40);(9,12,13,14,17,31,16,15)(10,19,20,29,22,21,32,37)(11,23,24,26,28,30,27,25)(18,33,34,38,36,35,39,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72286882, #72286947, #72286958, … |
| maximal normal subgroup of | #72286882, #72286947, #72286958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54168);