Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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×#385 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,16,9,15)(7,12,19,27,20,28,18,31)(8,21,24,30,25,32,23,13)(17,33,34,36,35,29,22,26);(2,4)(5,7,10,19,11,20,9,18)(6,12,14,27,16,28,15,31)(8,22,24,17,25,34,23,35)(13,29,21,26,30,33,32,36);(5,8,9,23,11,25,10,24)(6,13,14,21,16,30,15,32)(7,17,18,22,20,35,19,34)(12,26,27,33,28,36,31,29);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72252906, #72256886, #72256905 |
| maximal normal subgroup of | #72252906, #72256886, #72256905 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272592);