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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,13)(12,16,20,22)(18,23,24,21)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(1,2,5,7)(3,4,8,6)(10,15)(13,17)(16,22)(21,23)(25,27,30,33)(26,29,31,37)(28,34,35,39)(32,36,38,40);(1,3,5,8)(2,6,7,4)(12,20)(16,22)(18,24)(21,23)(26,31)(27,33)(28,35)(36,40);(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,24)(22,23)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,27)(26,29)(28,34)(30,33)(31,37)(32,36)(35,39)(38,40);(1,4)(2,3)(5,6)(7,8)(9,13)(10,11)(12,21)(14,17)(15,19)(16,18)(20,23)(22,24)(25,29)(26,27)(28,36)(30,37)(31,33)(32,34)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9156699);