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 | C23×C4 | C24 |
| prime radical | #35512 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,12,9,10,15,17)(7,11,14,18,13,16,19,20);(1,2)(3,4)(6,10)(8,15)(11,16)(14,19);(2,4)(11,16)(18,20);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)Polycyclic presentation:
2^9|a^2=h,h^2=i,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 | #72260927, #72260931, #72262275, … |
| maximal normal subgroup of | #72260927, #72260931, #72262275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287004);