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 | |
4388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,13,18,17)(21,22,23,24)(26,27)(31,33)(32,34)(38,39);(1,2,5,6)(3,7,8,4)(10,15)(11,20)(12,18)(22,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(2,6)(4,7)(10,15)(13,17)(16,19)(22,24)(29,37)(32,34)(35,40)(38,39);(1,3,5,8)(2,7,6,4)(9,11)(10,16)(12,23)(13,22)(14,20)(15,19)(17,24)(18,21)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,18)(10,17,15,13)(11,21,20,23)(16,22,19,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,17)(10,18,15,12)(11,22,20,24)(16,21,19,23)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8265470);