Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #95 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,11,13,12,14,15,16)(17,18)(19,27,29,23)(20,24)(21,26)(22,25)(28,32,30,31);(1,3)(2,5,6,7)(4,8)(17,19,21,30,22,29,20,28)(18,23,26,32,25,27,24,31);(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,e^2=i,f^2=g,b^a=bd,d^a=dg,f^a=fh,d^b=dh,e^b=eg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72309120, #72309121, #72309128, … |
| maximal normal subgroup of | #72309120, #72309121, #72309128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2875);