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 | |
4643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26)(27,33)(28,34)(29,32)(30,31)(35,37)(36,40)(38,39);(1,2)(3,7)(4,8)(5,6)(25,26)(27,28)(29,32)(30,31)(33,34)(35,36)(37,40)(38,39);(1,3,5,4)(2,7,6,8)(10,15)(11,16)(17,22)(18,23)(26,31)(27,34)(32,39)(35,40);(1,4,5,3)(2,8,6,7)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,39)(38,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28)(26,27)(29,37)(30,33)(31,34)(32,40)(35,39)(36,38);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,29,30,38)(26,32,31,39)(27,35,34,40)(28,36,33,37)Polycyclic presentation:
2^9|c^2=g,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7809429);