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 | |
4453 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)(3,8)(4,7)(5,6)(26,31)(27,33)(32,39)(34,36);(3,4)(7,8)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26)(27,29)(28,32)(30,31)(33,38)(34,37)(35,39)(36,40);(1,3)(2,7)(4,6)(5,8)(10,12)(11,17)(13,21)(22,24)(27,33)(28,35)(29,38)(32,39);(3,4)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27)(26,29)(28,36)(30,33)(31,38)(32,40)(34,35)(37,39);(1,4,6,3)(2,7,5,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40);(1,5,6,2)(3,7,4,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,27)(28,37)(30,38)(31,33)(32,34)(35,40)(36,39)Polycyclic presentation:
2^9|b^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8911120);