Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143957 |
| 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 | #19190 | 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 | |
384 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 384 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) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,3)(4,7,6,8)(9,10)(11,22)(12,21)(13,28)(14,20)(15,18)(16,23)(17,24)(19,31)(25,36)(26,34)(27,32)(29,38)(30,37)(33,39)(35,40);(2,7)(3,8)(4,6)(9,11)(10,16)(12,26)(13,24)(14,25)(15,23)(17,32)(18,36)(19,33)(20,34)(21,22)(27,40)(28,38)(29,31)(30,39)(35,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,23)(13,29)(14,25)(15,26)(16,32)(18,35)(19,34)(20,33)(21,38)(22,28)(24,31)(27,39)(30,40)(36,37)Polycyclic presentation:
2^9|a^2=gi,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,d^b=dg,f^b=fgi,d^c=dgi,e^c=eh,f^c=fgi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51175);