Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143821 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| 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 | |
346 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,7)(4,6)(5,9)(8,12)(10,15)(11,16)(13,14)(18,24)(19,28)(20,23)(21,35)(25,41)(26,36)(27,33)(29,47)(30,45)(31,37)(32,46)(34,38)(39,42)(43,44);(2,6)(4,12)(7,15)(8,9)(10,13)(14,16)(17,18)(19,29)(20,33)(21,36)(22,27)(23,24)(25,31)(26,38)(28,41)(30,43)(32,39)(34,40)(35,48)(37,42)(44,46)(45,47);(1,3,5,11)(2,7,9,16)(4,13,12,10)(6,14,8,15)(17,19,22,32)(18,25,27,43)(20,30,23,31)(21,37,38,45)(24,39,33,29)(26,44,36,41)(28,34,46,35)(40,42,48,47)Polycyclic presentation:
2^9|c^2=g,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=ei,f^b=fi,d^c=dgi,e^c=eh,f^c=fi,h^c=hi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51041);