Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C42×D4 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10)(11,20)(12,17)(13,27)(14,19)(15,18)(16,22)(21,34)(23,31)(24,37)(25,33)(26,32)(28,30)(29,35)(36,40)(38,39);(2,6)(3,7)(9,11,14,22,15,23,13,28)(10,16,19,31,18,30,27,20)(12,21,25,35,26,36,24,38)(17,29,33,40,32,39,37,34);(9,12,14,25,15,26,13,24)(10,17,19,33,18,32,27,37)(11,21,22,35,23,36,28,38)(16,29,31,40,30,39,20,34);(1,3,5,6,4,7,8,2)(10,18)(16,30)(17,32)(19,27)(20,31)(29,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=i,g^2=h,b^a=be,d^a=dhi,e^a=eh,d^b=di,e^b=eh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91225);