Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #106 |
| commutator | C8 | C82 |
| Frattini, cosocle | C42×C8 | C22 |
| socle | C22 | #1277 |
| prime radical | C23 | #341 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,19,14,21,30,18,16,22,31,38,15,20,13,32)(11,23,25,17,27,33,26,36,29,39,24,37,28,34,40,35);(2,5,7,6)(9,11,13,26,15,28,31,25,16,29,30,40,14,27,12,24)(10,17,19,34,21,36,18,23,22,37,38,33,20,35,32,39)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=ef,f^a=fi,h^a=hi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#1369, #72223664, C2×#93272 |
| maximal normal subgroup of | C8×#1369, #72223664, C2×#93272 |
| maximal quotient of | C2×#93272 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 468);