Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116201 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,18,13,19,14,24)(11,16,20,22,21,12,17,23);(1,2)(3,7,8,5)(4,6)(9,11)(10,16)(12,19)(13,21)(14,17)(15,20)(18,22)(23,24);(1,3)(2,5)(4,8)(6,7)(9,12,13,16)(10,17,19,20)(11,18,21,24)(14,23,15,22)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fhi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23400);