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 | #41166 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,18)(13,17)(14,19)(16,22)(20,24)(21,23)(25,26,30,31)(27,34,32,28)(29,33,37,38)(35,40,39,36);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,19,17)(12,20,22,21)(16,23,18,24)(26,31)(27,32)(33,38)(35,39);(1,2,5,4)(3,6,8,7)(9,12)(10,16)(11,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,27,30,32)(26,28,31,34)(29,35,37,39)(33,36,38,40);(9,13,14,11)(10,17,19,15)(12,21,22,20)(16,24,18,23)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,38)(37,40);(1,4,5,2)(3,7,8,6)(9,12)(10,18)(11,20)(13,21)(14,22)(15,24)(16,19)(17,23);(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,29,30,37)(26,33,31,38)(27,35,32,39)(28,36,34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7565088);