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 | |
4451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,17)(11,19,12,16)(13,14,18,24)(20,23,22,21)(26,31)(28,32)(33,39)(36,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,16,17,19)(13,20,18,22)(14,21,24,23)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,40)(38,39);(10,17)(14,24)(16,19)(21,23)(26,31)(28,32)(33,39)(36,40);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,16,17,19)(13,20,18,22)(14,23,24,21)(25,27,30,35)(26,32,31,28)(29,37,38,34)(33,36,39,40);(1,4,5,3)(2,8,6,7)(9,13,15,18)(10,14,17,24)(11,20,12,22)(16,21,19,23)(25,28,30,32)(26,27,31,35)(29,36,38,40)(33,34,39,37);(9,14)(10,18)(11,21)(12,23)(13,17)(15,24)(16,22)(19,20)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7984493);