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 | #46415 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,16)(12,19)(13,21)(14,15)(17,22)(18,20)(23,24)(26,29)(28,36)(31,34)(33,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26,30,29)(27,31,35,34)(28,32,36,37)(33,38,39,40);(2,5)(6,8)(10,15)(12,16)(17,21)(20,23)(25,27,30,35)(26,31,29,34)(28,33,36,39)(32,38,37,40);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,28)(26,32)(27,33)(29,37)(30,36)(31,38)(34,40)(35,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,29,30,26)(27,34,35,31)(28,37,36,32)(33,40,39,38)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=h,f^2=ghi,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, 8030159);