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 | |
4677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(7,8)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,39)(38,40);(1,2,6,5)(3,8,4,7)(12,19)(16,22)(18,23)(20,24)(25,27,30,34)(26,31,32,29)(28,35,37,40)(33,36,39,38);(1,2,6,5)(3,8,4,7)(10,15)(11,17)(16,22)(18,23)(26,32)(27,34)(33,39)(35,40);(1,3,6,4)(2,7,5,8)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,27)(26,29)(28,35)(30,34)(31,32)(33,38)(36,39)(37,40);(1,4,6,3)(2,7,5,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,36)(32,39)(34,40);(1,5,6,2)(3,8,4,7)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29)(26,27)(28,36)(30,31)(32,34)(33,40)(35,39)(37,38)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854410);