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 | |
4411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,13)(11,15,17,19)(12,20,22,16)(18,24,21,23)(25,26,31,29)(27,34,32,28)(30,33,38,37)(35,39,40,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,24)(21,22)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,39)(38,40);(1,3,4,7)(2,5,6,8)(11,17)(15,19)(18,21)(23,24)(26,29)(27,32)(30,38)(36,39);(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,21)(19,24)(25,28,31,34)(26,32,29,27)(30,39,38,36)(33,35,37,40);(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,29,31,26)(27,34,32,28)(30,37,38,33)(35,39,40,36);(9,13,14,10)(11,19,17,15)(12,16,22,20)(18,23,21,24)(25,30,31,38)(26,33,29,37)(27,35,32,40)(28,36,34,39)Polycyclic presentation:
2^9|a^2=ghi,c^2=gh,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7976538);