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 | #35807 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(12,20)(15,21)(18,24)(22,23)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,21)(13,17)(14,19)(15,20)(18,22)(23,24)(27,34)(30,33)(35,40)(37,39);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,27,29,34)(26,30,32,33)(28,35,36,40)(31,37,38,39);(1,4,5,3)(2,8,6,7)(9,12)(10,15)(11,18)(13,20)(14,22)(16,23)(17,21)(19,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,28,29,36)(26,31,32,38)(27,35,34,40)(30,37,33,39);(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=h,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, 7565808);