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 |
| prime radical | #34911 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4387 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,8)(5,7)(9,10,15,17)(11,18,21,16)(12,13,23,14)(19,22,24,20)(25,26,30,33)(27,29,35,31)(28,36,38,32)(34,37,40,39);(1,3,5,8)(2,6,7,4)(9,11)(10,16)(12,19)(13,20)(14,22)(15,21)(17,18)(23,24)(25,27,30,35)(26,31,33,29)(28,34,38,40)(32,37,36,39);(1,4,5,6)(2,3,7,8)(9,12)(10,13)(11,19)(14,17)(15,23)(16,20)(18,22)(21,24)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,39,31,37);(3,8)(4,6)(9,13,15,14)(10,12,17,23)(11,20,21,22)(16,19,18,24)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,40,36,34);(2,7)(3,8)(11,21)(16,18)(19,24)(20,22)(26,33)(27,35)(32,36)(34,40);(9,14,15,13)(10,12,17,23)(11,22,21,20)(16,19,18,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^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, 8012774);