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 | #36692 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(26,31)(27,35)(33,34)(36,37);(1,2)(3,6)(4,5)(7,8)(12,20)(15,22)(19,23)(21,24)(26,31)(32,38)(33,34)(39,40);(9,11,14,16)(10,13,17,18)(12,19,20,23)(15,21,22,24)(25,26,30,31)(27,32,35,38)(28,33,29,34)(36,39,37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,19)(13,21)(14,20)(16,23)(17,22)(18,24)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,39)(34,40);(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,22)(23,24)(25,28,30,29)(26,33,31,34)(27,36,35,37)(32,39,38,40);(9,13)(10,16)(11,17)(12,21)(14,18)(15,23)(19,22)(20,24)(25,29,30,28)(26,34,31,33)(27,37,35,36)(32,40,38,39)Polycyclic presentation:
2^9|c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7565630);