Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1982 |
| prime radical | C23 | #418 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
408 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(2,6,7,8)(9,10,14,20,13,17,30,36,15,21,31,38,23,19,12,18)(11,22,26,32,27,37,25,35,28,39,29,16,24,34,40,33);(1,2,3,6,5,7,4,8)(9,11,23,27,15,28,13,24)(10,16,19,33,21,32,17,35)(12,29,31,40,30,26,14,25)(18,37,38,39,36,34,20,22);(1,3,5,4)(2,6,7,8)(10,17,21,19)(11,24,28,27)(16,32)(18,20,36,38)(22,39)(25,26,40,29)(33,35)(34,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=egi,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ch,d^b=dg,f^b=fg,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48068);