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 | #45439 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,22)(21,24)(25,26,30,33)(27,29,36,39)(28,32,38,31)(34,40,35,37);(1,3,5,8)(2,6,7,4)(9,10,13,16)(11,14,18,17)(12,15,20,22)(19,21,24,23)(25,27,30,36)(26,29,33,39)(28,35,38,34)(31,40,32,37);(1,3)(2,4)(5,8)(6,7)(11,18)(12,20)(14,17)(15,22)(25,28,30,38)(26,31,33,32)(27,34,36,35)(29,37,39,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,33)(34,36)(39,40);(9,12)(10,15)(11,19)(13,20)(14,21)(16,22)(17,23)(18,24)(25,27)(26,29)(28,35)(30,36)(31,40)(32,37)(33,39)(34,38);(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,29)(26,27)(28,37)(30,39)(31,34)(32,35)(33,36)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669254);