Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C22×C4 | C2×C16 |
| Frattini, cosocle | C2×C4⋊C16 | C22 |
| socle | C22 | #1314 |
| prime radical | C23 | C22⋊C16 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C4⋊C16 ◁ G | |
| LECS | G ▷ C2×C4⋊C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 36 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 — 2 generators on 48 points:
(1,2)(3,9,11,16)(4,12,14,7)(5,8)(6,10)(13,15)(17,18,20,26,21,27,32,41,23,24,33,36,35,37,45,46,22,25,31,38,34,39,44,47,19,28,29,40,30,42,43,48);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,22,23)(18,24,25,28)(20,29,31,33)(21,30,34,35)(26,40,38,36)(27,37,39,42)(32,43,44,45)(41,48,47,46)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=gi,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,c^b=cgi,d^b=degi,d^c=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 892);