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 | |
4516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,16)(11,15,20,17)(12,13,19,18)(21,23,24,22)(25,26)(27,35)(28,38)(29,34)(30,32)(31,37)(33,39)(36,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,21)(13,23)(14,20)(16,17)(18,22)(19,24)(25,27,30,37)(26,31,32,35)(28,36,39,29)(33,40,38,34);(1,3,4,7)(2,6,5,8)(10,16)(11,20)(12,19)(22,23)(26,32)(31,35)(33,38)(34,40);(9,11)(10,17)(12,21)(13,22)(14,20)(15,16)(18,23)(19,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,40)(38,39);(9,12)(10,18)(11,21)(13,16)(14,19)(15,22)(17,23)(20,24)(25,28)(26,33)(27,36)(29,37)(30,39)(31,40)(32,38)(34,35);(9,13,14,18)(10,19,16,12)(11,22,20,23)(15,21,17,24)(25,29,30,36)(26,34,32,40)(27,28,37,39)(31,33,35,38)Polycyclic presentation:
2^9|a^2=i,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, 7882172);