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 | |
4561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,20)(14,15)(16,18)(17,21)(19,22)(23,24)(25,26,31,33)(27,35,38,32)(28,29,39,40)(30,34,37,36);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,19)(14,18)(15,16)(17,23)(20,22)(21,24)(25,27,31,38)(26,32,33,35)(28,36,39,34)(29,37,40,30);(1,4,5,2)(3,7,8,6)(10,15)(11,18)(12,21)(22,23)(26,33)(28,39)(32,35)(34,36);(9,11)(10,16)(12,19)(13,15)(14,18)(17,22)(20,23)(21,24)(25,28)(26,29)(27,36)(30,35)(31,39)(32,37)(33,40)(34,38);(9,12)(10,17)(11,19)(13,23)(14,21)(15,20)(16,22)(18,24)(25,29)(26,28)(27,37)(30,38)(31,40)(32,36)(33,39)(34,35);(9,13,14,16)(10,11,15,18)(12,22,21,23)(17,24,20,19)(25,30,31,37)(26,34,33,36)(27,29,38,40)(28,35,39,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,f^2=h,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, 7880126);