Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | C2×2−1+4 | C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,19,23,20,24,26,30,21,22,25,28,27,29,31,32);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(18,22)(19,25)(24,29)(26,31)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dg,e^b=eg,d^c=dg,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36869);