Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143142 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #2011 |
| prime radical | C23 | #459 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18,24,20)(19,30,33,43)(21,37,41,46)(22,29,23,28)(25,42,36,32)(26,40,27,39)(31,34,47,35)(38,45,48,44);(6,15)(9,16)(17,19,24,33)(18,25,20,36)(21,38,41,48)(22,35,23,34)(26,31,27,47)(28,45,29,44)(30,46,43,37)(32,40,42,39);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,20,24,18)(19,31,33,47)(21,37,41,46)(22,28,23,29)(25,38,36,48)(26,40,27,39)(30,35,43,34)(32,45,42,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=gi,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fg,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50377);