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 | |
4295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26)(27,36)(28,38)(29,40)(30,35)(31,32)(33,39)(34,37);(1,2)(3,6)(4,5)(7,8)(25,27,31,35)(26,30,32,36)(28,37,39,29)(33,34,38,40);(10,15)(11,16)(17,22)(18,23)(26,32)(27,35)(28,39)(34,40);(1,3,4,7)(2,6,5,8)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,28)(26,33)(27,37)(29,35)(30,34)(31,39)(32,38)(36,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29,31,37)(26,34,32,40)(27,28,35,39)(30,38,36,33);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,30)(26,35)(27,32)(28,34)(29,33)(31,36)(37,38)(39,40)Polycyclic presentation:
2^9|b^2=i,d^2=gh,e^2=i,f^2=hi,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, 7836669);