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 | |
4460 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)(11,15)(12,14)(13,20)(16,21)(17,18)(19,24)(22,23)(29,37)(33,38)(34,40)(36,39);(1,3,5,8)(2,6,4,7)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,38)(39,40);(10,12)(11,17)(16,20)(19,23)(26,31)(28,32)(33,38)(36,39);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,2,5,4)(3,6,8,7)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,28,30,32)(26,27,31,35)(29,36,37,39)(33,34,38,40);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621718);