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 | #42471 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,24)(21,23)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(3,7)(6,8)(9,11)(10,16)(12,14)(13,23)(15,20)(17,21)(18,22)(19,24)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,2,5,4)(3,6,7,8)(11,20)(14,21)(16,24)(18,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,3,5,7)(2,6,4,8)(9,12)(10,13)(11,21)(14,20)(15,17)(16,18)(19,22)(23,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,4,5,2)(3,8,7,6)(9,13)(10,17)(11,18)(12,19)(14,16)(15,22)(20,23)(21,24)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39);(9,14,15,21)(10,18,19,23)(11,12,20,17)(13,16,22,24)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325399);