Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19337 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
269 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16)(18,24)(19,28)(20,23)(21,35)(25,39)(26,43)(27,33)(29,41)(30,36)(31,45)(32,46)(34,37)(40,48)(42,44);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15)(17,18)(19,29)(20,33)(21,26)(22,27)(23,24)(25,31)(28,39)(30,42)(32,40)(34,47)(35,38)(36,41)(37,43)(44,46)(45,48);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)(17,19)(18,25,27,42)(20,31)(21,36)(22,32)(23,30)(24,29,33,40)(26,44,43,39)(28,35)(34,46)(37,45)(38,41,47,48)Polycyclic presentation:
2^9|c^2=dh,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fi,d^c=di,e^c=eghi,f^c=fgi,h^c=hi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52478);