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 | |
4563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26)(27,35)(28,37)(29,34)(30,31)(32,38)(33,39)(36,40);(1,2,5,6)(3,7,8,4)(10,11)(12,19)(16,18)(20,24)(26,31)(28,39)(32,35)(34,36);(2,6)(4,7)(13,21)(16,18)(20,24)(22,23)(26,31)(29,40)(32,35)(33,37);(2,6)(3,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27,30,38)(26,32,31,35)(28,34,39,36)(29,37,40,33);(1,3)(2,4)(5,8)(6,7)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,39)(26,33,31,37)(27,36,38,34)(29,32,40,35);(1,4)(2,3)(5,7)(6,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,40)(26,34,31,36)(27,37,38,33)(28,32,39,35)Polycyclic presentation:
2^9|b^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295987);