Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10,14,20,15,21,11,17)(12,18,23,16,13,19,22,24);(9,11,15,14)(10,16,21,24)(12,22,13,23)(17,19,20,18);(1,3,5,4)(2,6,8,7)(9,11,15,14)(10,17,21,20)(12,23,13,22)(16,19,24,18);(9,12,15,13)(10,18,21,19)(11,22,14,23)(16,17,24,20)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,e^b=ei,f^b=fgi,d^c=di,f^e=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387336);