Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8760 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,4,3,7)(9,10,13,19,14,18,12,17)(11,20,21,24,22,15,23,16);(1,3,6,5)(2,7,4,8)(9,11)(10,15)(12,21)(13,23)(14,22)(16,17)(18,20)(19,24);(1,4)(2,6)(3,8)(5,7)(9,11)(10,16,18,24)(12,21)(13,23)(14,22)(15,17,20,19);(9,12,14,13)(10,17,18,19)(11,21,22,23)(15,16,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=e,d^2=h,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,f^c=fi,g^c=gh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97551);