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 | #59228 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,16)(11,13)(12,21)(20,24)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,2,5,4)(3,7,8,6)(9,10)(11,17)(12,14)(13,23)(15,16)(18,21)(19,24)(20,22)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(2,4)(6,7)(9,11,15,13)(10,17,16,23)(12,19,21,22)(14,24,18,20)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(9,12)(10,18)(11,19)(13,22)(14,16)(15,21)(17,20)(23,24)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,3,5,8)(2,6,4,7)(9,13,15,11)(10,17,16,23)(12,22,21,19)(14,24,18,20)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,12,16,21)(11,20,13,24)(17,22,23,19)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=gh,e^2=hi,f^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, 8161898);