Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1521 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)(6,7)(9,10,13,17,14,12,21,16)(11,15,19,23,20,18,24,22);(5,6,8,7)(9,11,13,19,14,20,21,24)(10,15,17,23,12,18,16,22);(2,4)(5,7,8,6)(9,12)(10,14)(11,18)(13,16)(15,20)(17,21)(19,22)(23,24);(9,10)(11,18)(12,14)(13,17)(15,20)(16,21)(19,22)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,d^b=di,f^c=fi,g^c=gi,f^d=fi,g^d=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300056);