Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,14,11,15,9,17)(7,16,13,12,19,20,18,8);(5,7,10,13,11,19,9,18)(6,12,14,20,15,8,17,16);(5,8)(6,13)(7,17)(9,20)(10,16)(11,12)(14,19)(15,18);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,f^c=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72250020, #72257145, #72257175 |
| maximal normal subgroup of | #72250020, #72257145, #72257175 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299779);