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 | |
4328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,6)(3,5,8,4)(10,14)(12,20)(15,17)(18,24)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,37,39,36);(1,3)(2,4)(5,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,35)(26,31,33,28)(29,34,38,39)(32,37,40,36);(9,10,13,14)(11,15,19,17)(12,16,20,21)(18,22,24,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,37)(36,40)(38,39);(1,4,7,5)(2,8,6,3)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24);(1,5,7,4)(2,8,6,3)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,31)(26,27,33,35)(29,37,38,36)(32,39,40,34);(1,6,7,2)(3,5,8,4)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,37)(33,40)(35,39)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9754828);