Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2348 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)(5,6,11,15,9,14,10,16)(7,13,8,19,17,12,18,20);(2,4)(5,7)(6,12)(8,11)(9,17)(10,18)(13,14)(15,20)(16,19);(5,8,10,7,9,18,11,17)(6,13,15,19,14,12,16,20);(2,4)(6,14)(12,13)(15,16)(19,20)Polycyclic presentation:
2^9|a^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 | #72250537, #72256886, #72256917, … |
| maximal normal subgroup of | #72250537, #72256886, #72256917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267736);