Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,14,16,10)(8,11)(9,12)(13,17)(15,19,20,18);(1,2)(3,4)(5,7)(6,10)(8,15)(9,16)(11,18)(12,14)(13,19)(17,20);(2,4)(6,11)(8,17)(10,18)(12,13)(14,19)(15,20);(8,17)(11,13)(15,20)(18,19)Polycyclic presentation:
2^9|a^2=h,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219641, #72219668, #72219671, … |
| maximal normal subgroup of | #72219641, #72219668, #72219671, … |
| maximal quotient of | #72237989, #72256111 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287017);