Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483919 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33582 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(10,14,16,23)(11,17)(12,20)(15,19,21,24)(18,22);(1,2,4,5)(3,6,7,8)(10,15,16,21)(11,18)(14,19,23,24)(17,22);(2,5)(6,8)(9,10)(11,19)(12,21)(13,16)(14,22)(15,20)(17,23)(18,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,f^b=fi,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72264735, #72264817, #72264898, … |
| maximal normal subgroup of | #72264735, #72264817, #72264898, … |
| maximal quotient of | #72309592, #72309728, #72309815, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404172);