Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,19,21,23,20,24);(2,4)(17,19,22,23,21,24,20,18);(5,6,8,7)(10,13)(14,16)(17,20,21,22)(18,19,23,24);(5,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,19,23,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg,c^b=ci,d^b=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300763);