Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143438 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
387 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 387 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,6,10)(3,11,13,15)(4,7,5,9)(8,14,16,12)(17,18,24,30)(19,31,37,43)(20,38,22,42)(21,26,23,28)(25,35,46,33)(27,41,29,40)(32,45,39,44)(34,47,36,48);(2,7)(3,12)(4,5)(9,10)(11,15)(13,14)(17,19,24,37)(18,25,30,46)(20,39,22,32)(21,36,23,34)(26,44,28,45)(27,47,29,48)(31,42,43,38)(33,41,35,40);(1,3)(2,8)(4,12)(5,14)(6,13)(7,15)(9,11)(10,16)(18,26)(19,32)(20,40)(21,23)(22,41)(25,43)(27,29)(28,30)(31,46)(33,34)(35,36)(37,39)(44,48)(45,47)Polycyclic presentation:
2^9|a^2=gh,b^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=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fg,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50672);