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 | #54884 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4486 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)(3,5)(4,7)(6,8)(9,10)(11,19)(12,17)(13,16)(14,18)(15,22)(20,23)(21,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,38)(36,37)(39,40);(1,2,5,3)(4,7,8,6)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19)(25,27,30,35)(26,31,33,28)(29,37,39,34)(32,36,40,38);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,17)(13,16)(14,18)(19,22)(20,24)(21,23)(25,28,30,31)(26,27,33,35)(29,38,39,36)(32,37,40,34);(2,3)(6,7)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24)(25,29,30,39)(26,32,33,40)(27,34,35,37)(28,36,31,38);(1,4,5,8)(2,6,3,7)(9,12,14,13)(10,17,18,16)(11,21,22,20)(15,23,19,24)(26,33)(27,35)(32,40)(34,37);(9,13,14,12)(10,16,18,17)(11,21,22,20)(15,23,19,24)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=gh,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, 8012832);