Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #2537 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,20)(12,13)(14,19)(15,17)(16,21)(18,23)(22,24)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,22)(13,19)(14,18)(15,21)(17,20)(23,24)(27,33)(31,35)(34,39)(38,40);(1,2,5,4)(3,6,8,7)(10,17)(11,21)(12,18)(19,24)(25,27,30,33)(26,31,29,35)(28,34,36,39)(32,38,37,40);(9,12,15,18)(10,13,17,23)(11,22,21,14)(16,19,20,24);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,19)(12,17)(14,16)(15,23)(20,22)(21,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,14,15,22)(10,19,17,24)(11,18,21,12)(13,20,23,16)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7790197);