Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198144 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2083 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)(3,5,7,8)(4,6)(10,14,16,20)(11,17,19,18);(1,3)(2,5,6,8)(4,7)(9,10,12,20,13,16,21,14)(11,15,18,23,19,24,17,22);(9,11)(10,15)(12,18)(13,19)(14,22)(16,24)(17,21)(20,23);(3,7)(5,8)(9,12,13,21)(10,14,16,20)(11,18,19,17)(15,22,24,23)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=ghi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105347);