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 | #2537 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4274 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)(9,10,15,17)(11,21,22,16)(12,14,20,18)(13,23,24,19)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,40)(37,39);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,19)(13,18)(14,24)(15,22)(17,21)(20,23)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(10,17)(11,22)(12,20)(13,24)(25,26,30,34)(27,31,29,33)(28,32,38,36)(35,39,37,40);(9,12,15,20)(10,18,17,14)(11,19,22,23)(13,21,24,16)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(9,13,15,24)(10,19,17,23)(11,18,22,14)(12,21,20,16)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,20,17,12)(11,13,22,24)(16,19,21,23)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=hi,d^2=i,e^2=g,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, 7894490);