Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | #379 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7,6,3,4,8)(10,13,14,15)(18,19)(20,21)(23,24);(1,3)(2,6)(4,8)(5,7)(17,18,22,24)(19,20,23,21);(1,4,6,5)(2,7,3,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,20)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gh,d^2=h,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=ei,f^c=fgi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5962);