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 | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,5)(4,6,8,7)(9,10)(11,14)(12,19,21,15)(13,16)(17,24,20,22)(18,23);(2,5)(6,7)(9,11,13,18)(10,14,16,23)(12,20)(15,22)(17,21)(19,24);(2,6)(4,8)(5,7)(9,12)(10,15)(11,17)(13,21)(14,22)(16,19)(18,20)(23,24);(11,18)(14,23)(17,20)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=ei,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, 145465);