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 | |
457 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 98 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)(11,23)(12,27,29,32)(13,21)(14,17)(15,19)(16,22)(18,24,35,39)(20,26)(25,33)(28,38,30,37)(31,34,40,36);(2,5)(6,7)(9,11,15,25,16,26,13,14)(10,17,21,20,22,33,19,23)(12,24,30,40,29,39,28,31)(18,32,36,37,35,27,34,38);(1,3)(2,6)(4,8)(5,7)(9,12)(10,18)(11,24)(13,28)(14,31)(15,30)(16,29)(17,32)(19,34)(20,37)(21,36)(22,35)(23,38)(25,40)(26,39)(27,33)Polycyclic presentation:
2^9|a^2=f,b^2=dghi,d^2=i,e^2=hi,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.
| maximal subgroup of | #72265511, #72265529, #72265692, … |
| maximal normal subgroup of | #72265511, #72265529, #72265692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41743);