Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,21)(13,16)(14,19)(15,22)(18,23)(20,24);(2,5)(6,8)(9,10)(11,18)(12,15)(13,16)(14,20)(17,23)(19,24)(21,22);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14,16,18)(12,19)(13,20)(15,17,21,24)(22,23);(3,7)(6,8)Polycyclic presentation:
2^9|c^2=h,f^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,c^b=cg,h^b=hi,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311609, #72311619, #72311801, … |
| maximal normal subgroup of | #72311609, #72311619, #72311801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289613);