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 | #45817 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4192 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)(3,4)(5,6)(7,8)(9,10)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,40)(38,39);(2,6)(3,7)(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,23)(22,24)(25,27)(26,32)(28,38)(29,35)(30,39)(31,40)(33,34)(36,37);(1,3,5,7)(2,4,6,8)(10,15)(11,16)(17,21)(19,24)(25,28,31,35)(26,33,36,30)(27,38,40,29)(32,34,37,39);(1,4)(2,7)(3,6)(5,8)(9,11,14,16)(10,12,15,18)(13,19,23,24)(17,20,21,22);(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,29,31,38)(26,34,36,39)(27,35,40,28)(30,32,33,37);(9,13)(10,17)(11,19)(12,20)(14,23)(15,21)(16,24)(18,22)(25,30,31,33)(26,35,36,28)(27,39,40,34)(29,37,38,32)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9624807);