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 | |
4442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,19)(12,17)(13,23)(14,16)(15,20)(18,21)(22,24);(1,2)(3,6)(4,7)(5,8)(9,11,14,20)(10,15,16,19)(12,21,22,13)(17,23,24,18)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(1,3,5,4)(2,6,8,7)(10,16)(11,20)(12,22)(18,23)(26,30)(31,38)(32,35)(39,40);(9,12)(10,17)(11,21)(13,20)(14,22)(15,23)(16,24)(18,19)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,3,5,4)(2,6,8,7)(9,13,14,21)(10,18,16,23)(11,22,20,12)(15,24,19,17)(25,28)(26,32)(27,33)(29,37)(30,35)(31,39)(34,36)(38,40);(1,4,5,3)(2,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,20)(21,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7698246);