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 | #59110 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4444 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)(11,15)(12,17)(13,18)(14,19)(16,20)(21,23)(22,24)(25,26)(27,32)(28,31)(29,37)(30,34)(33,39)(35,36)(38,40);(2,6)(3,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,19)(17,23)(18,24)(25,27,31,35)(26,32,28,36)(29,34,39,38)(30,33,40,37);(1,3,5,7)(2,4,6,8)(11,20)(15,16)(21,24)(22,23)(26,28)(27,35)(33,37)(34,38);(1,4)(2,7)(3,6)(5,8)(9,11)(10,16)(12,21)(13,23)(14,20)(15,19)(17,22)(18,24)(25,28,31,26)(27,32,35,36)(29,37,39,33)(30,34,40,38);(9,12)(10,17)(11,21)(13,19)(14,18)(15,23)(16,22)(20,24)(25,29)(26,33)(27,34)(28,37)(30,36)(31,39)(32,40)(35,38);(9,13,14,17)(10,18,19,12)(11,22,20,23)(15,24,16,21)(25,30,31,40)(26,34,28,38)(27,33,35,37)(29,32,39,36)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653568);