Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8359 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,4,6)(5,8)(10,14,15,16)(17,18,19,21,20,22,23,24);(1,3,5,4)(2,6,8,7)(9,10,13,15)(11,14,12,16);(1,4,5,3)(2,7,8,6)(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=gh,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,f^b=fgh,f^c=fg86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4785);