Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134444 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1802 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,6)(4,5)(9,10,13,21,16,22,15,19)(11,20,14,17,25,31,24,32)(12,26,27,39,28,40,30,36)(18,29,34,37,35,38,33,23);(2,5)(6,7)(9,11,15,24,16,25,13,14)(10,17,21,31,22,32,19,20)(12,23,30,38,28,37,27,29)(18,26,34,39,35,40,33,36);(1,3)(2,6)(4,8)(5,7)(9,12)(10,18)(11,23)(13,27)(14,29)(15,30)(16,28)(17,26)(19,33)(20,36)(21,34)(22,35)(24,38)(25,37)(31,39)(32,40)Polycyclic presentation:
2^9|a^2=f,b^2=dgh,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^c=eh,f^c=fghi,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, 41744);