Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238267 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,14)(12,19,21,15)(13,16)(17,24,20,22)(18,23);(9,11)(10,14)(12,20,21,17)(13,18)(15,22,19,24)(16,23);(1,2,4,6)(3,8,7,5)(9,12,13,21)(10,15,16,19)(11,17,18,20)(14,22,23,24);(11,18)(14,23)(17,20)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=i,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,d^b=dh,g^b=gh,e^c=eh,f^c=fhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145467);