Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,10,13,11,14,9,8)(7,15,18,12,17,16,19,20);(1,2,4,3)(5,6,10,13,11,14,9,8)(7,16,19,12,17,15,18,20);(5,7)(6,12)(8,15)(9,19)(10,18)(11,17)(13,16)(14,20);(1,3,4,2)(5,8,9,14,11,13,10,6)(7,15,19,20,17,16,18,12)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72252681, #72252977, #72256917, … |
| maximal normal subgroup of | #72252681, #72252977, #72256917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278353);