Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| 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 | |
306 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 306 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)(17,18,22,28,24,26,41,43,23,27,40,45,21,38,30,47)(19,29,33,39,34,44,48,20,32,36,46,35,31,25,42,37);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,19,30,42,21,31,40,46,23,32,41,48,24,34,22,33)(18,25,43,39,38,36,28,37,27,44,47,35,26,29,45,20);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)(17,20)(18,19)(21,39)(22,29)(23,37)(24,35)(25,30)(26,34)(27,32)(28,46)(31,38)(33,45)(36,40)(41,44)(42,43)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dgi,e^b=eg,f^b=fgi,d^c=dg,e^c=egi,f^c=fg74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45064);