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 | |
4389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,5,4)(9,10)(11,16)(12,22)(13,17)(14,18)(15,21)(19,20)(23,24)(25,26,30,34)(27,31,35,38)(28,32,37,40)(29,33,39,36);(1,3,6,5)(2,7,8,4)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,23)(22,24)(25,27)(26,31)(28,36)(29,32)(30,35)(33,37)(34,38)(39,40);(2,8)(3,5)(12,17)(13,22)(19,23)(20,24)(27,35)(28,37)(31,38)(32,40);(1,3,6,5)(2,7,8,4)(9,12,14,17)(10,13,18,22)(11,19,21,23)(15,20,16,24)(25,28)(26,32)(27,36)(29,31)(30,37)(33,35)(34,40)(38,39);(1,4,6,7)(2,3,8,5)(9,11)(10,16)(12,23)(13,20)(14,21)(15,18)(17,19)(22,24)(25,29)(26,33)(27,32)(28,38)(30,39)(31,37)(34,36)(35,40);(1,5,6,3)(2,7,8,4)(9,13)(10,17)(11,20)(12,18)(14,22)(15,23)(16,19)(21,24)(25,27)(26,31)(28,36)(29,32)(30,35)(33,37)(34,38)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8815180);