Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | C2×#135 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,21,24,22,25,29,31,20,19,23,26,28,27,30,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,19)(24,26)(25,27)(31,32);(2,8)(7,15)(9,13)(10,16)(17,19,21,26,22,27,29,32,20,18,23,24,28,25,30,31);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(18,19)(24,26)(25,27)(31,32)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=g,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,c^b=ci,e^b=eg,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390805);