Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
291 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10,14,20)(11,22,25,38)(12,16,30,19)(13,31,27,18)(15,32,29,21)(17,28,36,24)(23,37,40,39)(26,35,33,34);(1,3,5,6,4,7,8,2)(10,16)(12,27)(13,30)(15,29)(17,34)(18,21)(19,20)(22,37)(23,24)(26,33)(28,40)(31,32)(35,36)(38,39);(2,6)(3,7)(9,11)(10,17)(12,28)(13,23)(14,25)(15,33)(16,34)(18,38)(19,35)(20,36)(21,39)(22,31)(24,30)(26,29)(27,40)(32,37)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ch,d^b=dg,e^b=egi,g^b=gi,d^c=dgi,e^c=egi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43123);