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 | |
4250 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)(3,7)(4,5)(6,8)(10,11)(15,17)(16,18)(22,23)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(2,5)(6,7)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(27,36)(29,37)(32,35)(34,40);(1,2)(3,6)(4,5)(7,8)(12,19)(13,21)(15,17)(16,18)(25,27,31,36)(26,32,30,35)(28,29,39,37)(33,40,38,34);(1,3,4,8)(2,6,5,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,31,39)(26,33,30,38)(27,37,36,29)(32,34,35,40);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,31,37)(26,34,30,40)(27,28,36,39)(32,38,35,33);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,26)(27,32,36,35)(28,38,39,33)(29,40,37,34)Polycyclic presentation:
2^9|b^2=ghi,c^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=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9397280);