Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C2×#537 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5,6,7)(9,10,13,18,14,17,28,32)(11,19,22,33,21,15,26,30)(12,23,25,34,24,16,27,31)(20,35,37,40,36,29,38,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,20)(13,26)(14,21)(16,29)(17,19)(18,33)(22,28)(23,35)(24,36)(25,38)(27,37)(30,32)(31,40)(34,39);(9,12,14,24)(10,16,17,23)(11,20,21,36)(13,27,28,25)(15,29,19,35)(18,34,32,31)(22,38,26,37)(30,40,33,39);(10,17)(15,19)(16,23)(18,32)(29,35)(30,33)(31,34)(39,40)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,g^c=gh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93083);