Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,7,14,12)(8,11,17,19,16,15,20,18);(5,7)(6,10)(8,15)(9,12)(11,16)(13,14)(17,18)(19,20);(2,4);(5,8,9,17,10,16,14,20)(6,11,13,19,7,15,12,18);(8,16)(11,15)(17,20)(18,19)Polycyclic presentation:
2^9|a^2=h,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72320787 |
| maximal normal subgroup of | #72320787 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279888);