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 | #49561 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26,31,33)(27,32,37,38)(28,34,35,29)(30,36,39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24)(25,27,31,37)(26,32,33,38)(28,36,35,40)(29,30,34,39);(11,20)(13,21)(15,19)(17,24)(26,33)(27,37)(28,35)(30,39);(9,12)(10,16)(11,21)(13,20)(14,23)(15,17)(18,22)(19,24)(25,28)(26,34)(27,36)(29,33)(30,38)(31,35)(32,39)(37,40);(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,29,31,34)(26,35,33,28)(27,30,37,39)(32,40,38,36);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,17,18,24)(11,12,20,23)(15,22,19,16)(25,30)(26,36)(27,34)(28,38)(29,37)(31,39)(32,35)(33,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7673781);