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 | |
4505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,16)(12,21)(13,15)(14,18)(17,22)(19,20)(23,24)(25,26,31,35)(27,36,39,32)(28,29,40,33)(30,34,38,37);(1,2)(3,6)(4,7)(5,8)(9,11,15,19)(10,16,13,20)(12,14,22,24)(17,23,21,18)(25,27,31,39)(26,32,35,36)(28,37,40,34)(29,38,33,30);(10,13)(11,19)(12,22)(18,23)(25,28,31,40)(26,33,35,29)(27,37,39,34)(30,36,38,32);(9,12)(10,17)(11,14)(13,21)(15,22)(16,23)(18,20)(19,24)(25,28)(26,29)(27,37)(30,36)(31,40)(32,38)(33,35)(34,39);(1,3,5,4)(2,6,8,7)(9,13,15,10)(11,20,19,16)(12,21,22,17)(14,18,24,23)(25,29)(26,28)(27,38)(30,39)(31,33)(32,37)(34,36)(35,40);(1,4,5,3)(2,7,8,6)(9,14)(10,18)(11,12)(13,23)(15,24)(16,21)(17,20)(19,22)(25,30,31,38)(26,34,35,37)(27,29,39,33)(28,36,40,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7879706);