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 | #121 | M16 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 289 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:
(1,2,5,9,6,10,4,8)(3,11,13,7,14,15,12,16)(17,18,22,28,23,26,40,43,24,27,41,45,21,38,30,47)(19,29,33,39,34,44,48,20,32,36,46,35,31,25,42,37);(1,3)(2,7)(4,13)(5,12)(6,14)(8,15)(9,11)(10,16)(17,19,30,42,21,31,41,46,24,32,40,48,23,34,22,33)(18,25,43,39,38,36,28,37,27,44,47,35,26,29,45,20);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,15)(10,16)(17,20)(18,19)(21,39)(22,29)(23,35)(24,37)(25,30)(26,34)(27,32)(28,46)(31,38)(33,45)(36,41)(40,44)(42,43)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,d^b=dg,e^b=ei,f^b=fg,d^c=di,e^c=eg,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43950);