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 | |
4926 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)(3,4)(5,6)(7,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,22)(18,23)(21,24);(2,6)(3,7)(9,11,13,19)(10,14,15,17)(12,21,22,18)(16,24,20,23)(25,26,30,31)(27,29,33,38)(28,35,37,32)(34,40,39,36);(1,3,5,7)(2,4,6,8)(10,15)(14,17)(16,20)(23,24)(26,31)(29,38)(32,35)(36,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,24)(21,22)(25,27,30,33)(26,29,31,38)(28,34,37,39)(32,36,35,40);(9,12,13,22)(10,16,15,20)(11,18,19,21)(14,23,17,24)(25,28,30,37)(26,32,31,35)(27,34,33,39)(29,36,38,40);(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,29)(26,33)(27,31)(28,36)(30,38)(32,39)(34,35)(37,40)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=ghi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8134607);