Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134985 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1804 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
297 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(2,5)(3,7)(6,8)(9,10,12,18,15,20,14,32)(11,21,24,16,25,35,22,36)(13,27,26,33,31,40,30,19)(17,29,38,28,34,39,37,23);(1,2,3,5,4,6,7,8)(9,11)(10,16,20,36)(12,22)(13,28)(14,24)(15,25)(17,33,34,19)(18,21,32,35)(23,31)(26,29)(27,37,40,38)(30,39);(9,11)(10,17)(12,24)(13,29)(14,22)(15,25)(16,33)(18,38)(19,36)(20,34)(21,27)(23,30)(26,28)(31,39)(32,37)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=dh,d^2=gi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42183);