Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1850 |
| prime radical | C23 | #424 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
243 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,22,26,21,28,38,31,23,27,39,19,29,25,43,32)(20,33,35,46,37,42,48,30,36,44,47,45,34,24,40,41);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19,22,25,21,32,38,18,23,26,39,28,29,31,43,27)(20,30,35,44,37,45,48,24,36,41,47,33,34,46,40,42);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,20,23,36)(18,24,27,42)(19,30,26,41)(21,37,29,34)(22,35,39,47)(25,44,28,33)(31,46,32,45)(38,48,43,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=i,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,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43059);