Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214664 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #35346 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2840 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)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(2,4)(11,18)(17,19);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)Polycyclic presentation:
2^9|a^2=h,b^a=be,c^a=cf,e^a=ei,c^b=cg,h^b=hi,d^c=de,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225631, #72225638, #72225641, … |
| maximal normal subgroup of | #72225631, #72225638, #72225641, … |
| maximal quotient of | #72234623, #72237884, #72240400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287002);