Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1850 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 104 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)(9,10,14,20,16,19,32,36,15,21,31,37,13,23,30,38)(11,22,25,28,18,17,33,29,26,34,40,27,24,35,39,12);(2,6)(3,4)(7,8)(9,11,14,25,16,18,32,33,15,26,31,40,13,24,30,39)(10,17,20,29,19,34,36,27,21,35,37,12,23,22,38,28);(2,7)(3,4)(6,8)(9,12)(10,18)(11,23)(13,27)(14,22)(15,29)(16,28)(17,32)(19,26)(20,33)(21,24)(25,38)(30,35)(31,34)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=egi,d^c=dgi,e^c=eg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43075);