Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
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 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(6,8)(9,10,14,18,13,16,12,17)(11,19,22,15,21,24,20,23);(1,2,3,4)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,24)(18,19);(5,6,7,8)(10,16)(15,23)(17,18)(19,24);(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,e^b=ei,h^b=hi,g^d=gi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300285);