Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,16,13,12,20,21)(11,17,19,24,18,15,22,23);(9,11,14,22,13,18,20,19)(10,15,21,23,12,17,16,24);(2,4)(9,12)(10,13)(11,17)(14,16)(15,18)(19,23)(20,21)(22,24);(6,8)(11,18)(15,17)(19,22)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,d^b=di,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299889);