Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #189517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2283 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,20,23,16)(12,24)(13,15)(14,19)(17,25)(18,27)(21,34,36,38)(22,31,37,28)(26,29)(30,35,33,40)(32,39);(1,3)(2,5)(4,8)(6,7)(10,15,19,18)(16,28,20,31)(17,29,32,24)(30,34,33,38);(3,8)(5,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,28)(17,30)(18,31)(19,20)(24,38)(25,35)(26,36)(27,37)(29,34)(32,33)(39,40);(2,6)(5,7)(9,12,13,25,14,26,27,39)(10,17,18,24,19,32,15,29)(11,21,22,35,23,36,37,40)(16,30,31,38,20,33,28,34)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270844);