Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9431 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #3171 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
310 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2)(3,6,10,16)(4,9)(5,8)(7,11,15,12)(13,14)(17,18,22,28,23,29,41,47,24,27,42,46,21,26,31,44)(19,30,34,39,32,45,48,20,33,36,40,35,38,25,43,37);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,19,31,48,21,38,42,34,24,33,41,43,23,32,22,40)(18,25,44,39,26,36,46,37,27,45,47,35,29,30,28,20);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)(17,20)(18,19)(21,39)(22,30)(23,35)(24,37)(25,31)(26,38)(27,33)(28,40)(29,32)(34,46)(36,42)(41,45)(43,47)(44,48)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dgi,e^b=egi,f^b=fgi,d^c=dgi,e^c=eg,f^c=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45067);