Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#472 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,7,8)(9,10,12,20,14,21,27,16,15,22,30,18,13,19,29,33)(11,23,24,39,25,37,40,35,26,38,28,32,31,17,34,36);(1,2,6,5,4,7,3,8)(10,16,22,33)(12,27)(13,14)(17,32,37,39)(18,21,20,19)(23,35,38,36)(24,40)(25,31)(28,34)(29,30);(9,11)(10,17)(12,28)(13,31)(14,25)(15,26)(16,32)(18,36)(19,38)(20,35)(21,23)(22,37)(24,30)(27,34)(29,40)(33,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216438, #72216443, #72216456, … |
| maximal normal subgroup of | #72216438, #72216443, #72216456, … |
| maximal quotient of | #72216588, C2×#136616 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43812);