Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145505 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2054 |
| 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 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 272 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10,15,20)(11,21,26,35)(12,16,29,33)(13,30,27,18)(14,31,28,19)(17,23,36,24)(22,37,39,34)(25,32,40,38);(1,2)(3,7)(4,6)(5,8)(10,16)(11,22)(12,27)(13,29)(14,28)(17,34)(18,19)(20,33)(21,35)(23,40)(24,25)(26,39)(30,31)(36,37);(2,6)(3,4)(7,8)(9,11,12,23,28,40,13,22,15,26,29,24,14,25,27,39)(10,17,18,35,31,37,33,38,20,36,30,21,19,34,16,32)Polycyclic presentation:
2^9|a^2=i,c^2=d,d^2=gi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fhi,g^b=gi,h^b=hi,e^c=egh,f^c=fghi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52702);