Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,14,8,17)(7,11,15,12,16,20,18,19);(2,4)(5,7)(6,11)(8,18)(9,15)(10,16)(12,13)(14,20)(17,19);(5,7)(6,12)(8,18)(9,15)(10,16)(11,17)(13,20)(14,19);(2,4)Polycyclic presentation:
2^9|a^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72258084, #72258101, #72261217, … |
| maximal normal subgroup of | #72258084, #72258101, #72261217, … |
| maximal quotient of | #72261905, #72303522, #72303872 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266666);