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 | |
4435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,16)(12,19)(13,21)(14,15)(17,22)(18,20)(23,24)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(3,7)(6,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(26,28)(27,33)(32,36)(35,39);(1,2,5,4)(3,6,7,8)(10,15)(12,16)(17,21)(20,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3,5,7)(2,6,4,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,4,5,2)(3,8,7,6)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|c^2=hi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026479);