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 | |
4534 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26,31,32)(27,37,39,33)(28,35,29,34)(30,36,40,38);(1,2,4,6)(3,5,8,7)(10,15)(12,16)(17,22)(20,23)(26,32)(28,29)(33,37)(36,38);(1,2)(3,5)(4,6)(7,8)(13,21)(17,22)(18,24)(20,23)(25,27,31,39)(26,33,32,37)(28,36,29,38)(30,34,40,35);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28,31,29)(26,34,32,35)(27,36,39,38)(30,33,40,37);(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,29,31,28)(26,35,32,34)(27,38,39,36)(30,37,40,33);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30)(26,36)(27,35)(28,33)(29,37)(31,40)(32,38)(34,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8134657);