Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145480 |
| 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 | #33779 | C2 |
| 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 | |
295 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 295 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)(3,5)(4,7)(6,8)(10,18)(11,25)(12,31)(13,14)(15,16)(20,23)(21,24)(22,32)(26,34)(27,29)(28,33)(30,39)(35,36)(38,40);(3,8)(5,6)(9,10)(11,26)(12,20)(13,21)(14,32)(15,23)(16,22)(17,24)(18,31)(19,27)(25,35)(28,40)(29,34)(30,38)(33,37)(36,39);(2,5)(3,8)(6,7)(9,11,12,27,16,29,13,25,17,30,14,33,15,28,31,39)(10,19,18,26,23,36,32,40,24,37,21,38,22,35,20,34)Polycyclic presentation:
2^9|c^2=d,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,h^b=hi,e^c=egh,f^c=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52676);