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 | #2661 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,21)(12,18)(13,24)(14,20)(15,17)(16,23)(19,22)(25,26,32,36)(27,37,38,33)(28,34,29,39)(30,31,35,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,18)(14,22)(15,23)(17,21)(20,24)(25,27,32,38)(26,33,36,37)(28,35,29,30)(31,39,40,34);(10,17)(12,20)(16,21)(19,24)(25,28,32,29)(26,34,36,39)(27,30,38,35)(31,33,40,37);(1,3,5,4)(2,6,8,7)(9,12,15,20)(10,18,17,14)(11,19,23,24)(13,21,22,16)(25,29,32,28)(26,34,36,39)(27,30,38,35)(31,37,40,33);(9,13,15,22)(10,19,17,24)(11,18,23,14)(12,21,20,16)(25,30)(26,31)(27,28)(29,38)(32,35)(33,39)(34,37)(36,40);(1,4,5,3)(2,7,8,6)(9,14,15,18)(10,20,17,12)(11,22,23,13)(16,24,21,19)(25,31)(26,35)(27,34)(28,33)(29,37)(30,36)(32,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=i,d^2=g,e^2=h,f^2=gi,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, 7893396);