Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10,15,17)(11,19,22,16)(12,13,18,14)(20,21,24,23)(25,26)(27,33)(28,30)(29,36)(31,34)(32,38)(35,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,20)(13,23)(14,21)(15,22)(17,19)(18,24)(25,27)(26,31)(28,33)(29,37)(30,34)(32,40)(35,38)(36,39);(1,4,5,6)(2,7,8,3)(9,12)(10,13)(11,20)(14,17)(15,18)(16,23)(19,21)(22,24)(25,28,30,26)(27,33,34,31)(29,36,38,32)(35,40,37,39);(3,7)(4,6)(10,17)(12,18)(16,19)(20,24)(26,28)(27,34)(32,36)(35,37);(2,8)(3,7)(9,13,15,14)(10,18,17,12)(11,21,22,23)(16,20,19,24)(25,29,30,38)(26,32,28,36)(27,35,34,37)(31,39,33,40);(9,14,15,13)(10,12,17,18)(11,21,22,23)(16,20,19,24)Polycyclic presentation:
2^9|a^2=gh,c^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012768);