Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188770 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2285 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,13,19,14,17,12,18)(11,20,21,24,22,15,23,16);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,23)(14,22)(16,18)(17,20)(19,24);(2,5)(6,8)(9,11)(10,16,17,24)(12,21)(13,23)(14,22)(15,18,20,19);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,21)(13,23)(14,22)(16,18)(17,20)(19,24)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270568);