Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342612 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,17,14,16,21,11)(12,15,20,23,19,22,24,18);(2,5)(6,7)(9,11,13,16,14,17,21,10)(12,18,20,22,19,23,24,15);(1,3,4,8)(2,6,5,7);(9,12)(10,15)(11,18)(13,20)(14,19)(16,22)(17,23)(21,24);(12,19)(15,22)(18,23)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,h^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=hi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279430);