Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,10)(8,15,16,11)(9,12)(13,19,20,17)(14,18);(1,2)(3,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,18)(15,19)(16,20);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,20)(18,19);(2,4);(7,14)(10,18)(13,20)(17,19)Polycyclic presentation:
2^9|a^2=h,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225617, #72225620, #72225623, … |
| maximal normal subgroup of | #72225617, #72225620, #72225623, … |
| maximal quotient of | #72240560, #72249356, C2×#6379010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286206);