Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1580 |
| prime radical | C23 | #444 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
205 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,6)(4,5)(7,8)(9,10,14,19,13,20,23,18,15,21,30,38,17,12,28,29)(11,22,26,33,24,16,31,34,27,35,25,37,32,36,40,39);(2,5)(6,8)(9,11,23,31,17,32,14,26,15,27,28,40,13,24,30,25)(10,16,29,33,12,22,38,39,21,36,18,37,20,35,19,34);(1,3,4,7)(2,6,5,8)(9,12,15,20)(10,17,21,13)(11,22,27,35)(14,18,30,29)(16,32,36,24)(19,23,38,28)(25,33,26,37)(31,39,40,34)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=dg,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,e^b=ei,d^c=di,e^c=egi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47281);