Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1893 |
| prime radical | C2×#472 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(1,2,5,7,4,3,6,8)(9,10,14,19,16,12,15,27)(11,20,23,37,24,36,32,38)(13,28,30,26,21,39,31,18)(17,33,34,22,25,40,35,29);(1,3,5,8,4,2,6,7)(9,11,21,22,15,23,30,40,16,24,13,29,14,32,31,33)(10,17,18,36,19,35,28,37,12,25,26,20,27,34,39,38);(2,3)(7,8)(9,12)(10,16)(11,17)(13,18)(14,19)(15,27)(20,29)(21,26)(22,36)(23,35)(24,25)(28,31)(30,39)(32,34)(33,38)(37,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44854);