Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131820 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #61284 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
597 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 106 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:
(2,3)(4,5)(7,8)(9,10,15,22,17,21,13,32,16,23,34,40,14,35,33,20)(11,24,27,18,26,37,25,39,12,28,31,38,30,19,29,36);(1,2,6,8)(3,4,7,5)(9,11,16,12)(10,18,23,38)(13,31,33,27)(14,26,17,30)(15,29,34,25)(19,20,37,32)(21,36,35,39)(22,24,40,28);(1,3)(2,4)(5,8)(6,7)(9,12,16,11)(10,19,23,37)(13,29,33,25)(14,26,17,30)(15,31,34,27)(18,20,38,32)(21,24,35,28)(22,36,40,39)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=hi,e^2=g,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=eg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39120);