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 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,30,32)(27,35,36,31)(28,39,38,33)(29,34,37,40);(1,2,5,6)(3,7,4,8)(25,27,30,36)(26,31,32,35)(28,37,38,29)(33,40,39,34);(1,3,5,4)(2,7,6,8)(10,15)(11,16)(17,22)(18,23)(26,32)(27,36)(28,38)(34,40);(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,40)(38,39);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28)(26,33)(27,37)(29,36)(30,38)(31,40)(32,39)(34,35);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,29,30,37)(26,34,32,40)(27,38,36,28)(31,39,35,33)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=hi,f^2=ghi,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, 7894518);