Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2356 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)(5,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,23)(22,24);(9,11,13,18)(10,14,16,21)(12,17,19,24)(15,22,23,20);(2,4)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22);(5,7,8,6)(11,18)(14,21)(17,24)(20,22)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266223);