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 | |
4493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,18,24,21)(26,30)(31,37)(32,35)(38,40);(1,3,5,8)(2,6,7,4)(10,15)(11,20)(17,22)(18,21)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,3)(2,4)(5,8)(6,7)(10,15)(12,23)(16,19)(18,21)(27,33)(31,37)(34,39)(38,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,23)(22,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(9,12)(10,17)(11,21)(13,16)(14,23)(15,22)(18,20)(19,24)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39);(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^9|a^2=hi,b^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8458421);