Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1845 |
| prime radical | C23 | #494 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,6)(7,8)(9,10,13,18,15,19,14,29)(11,20,23,35,25,36,24,39)(12,26,27,33,28,40,21,17)(16,30,31,38,32,37,34,22);(9,11,21,37,14,24,28,38,15,25,27,30,13,23,12,22)(10,16,26,35,18,31,33,36,19,32,40,39,29,34,17,20);(1,2,6,8,4,7,3,5)Polycyclic presentation:
2^9|a^2=f,b^2=dg,c^2=eh,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi140 conjugacy classes — view the full character table.
| maximal subgroup of | #72216239, #72216249, #72216251, … |
| maximal normal subgroup of | #72216239, #72216249, #72216251, … |
| maximal quotient of | #72216239 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42977);