Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2088 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)(11,15)(12,14)(13,17)(16,20)(18,24)(19,23)(21,22);(10,14)(12,20)(15,23)(16,17)(19,21)(22,24);(2,4)(9,11,13,18)(10,15,17,24)(12,21,20,19)(14,22,16,23);(5,7,8,6)(11,18)(15,24)(19,21)(22,23)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266260);