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 | #48611 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,13)(14,18)(15,21)(16,22)(19,23)(20,24)(25,26)(27,36)(28,38)(29,35)(30,32)(31,33)(34,39)(37,40);(1,2,5,8)(3,4,7,6)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,31,30)(26,32,33,36)(28,37,39,35)(29,34,40,38);(1,3)(2,6)(4,8)(5,7)(26,33)(28,39)(32,36)(35,37);(1,4)(2,7)(3,8)(5,6)(9,12,14,22)(10,16,18,13)(11,19,21,24)(15,23,17,20)(25,28)(26,34)(27,37)(29,36)(30,35)(31,39)(32,40)(33,38);(1,3)(2,4)(5,7)(6,8)(9,13,14,16)(10,12,18,22)(11,20,21,23)(15,19,17,24)(25,29,31,40)(26,35,33,37)(27,38,30,34)(28,32,39,36);(1,4)(2,7)(3,8)(5,6)(9,11)(10,17)(12,19)(13,23)(14,21)(15,18)(16,20)(22,24)(25,30,31,27)(26,36,33,32)(28,35,39,37)(29,38,40,34)Polycyclic presentation:
2^9|b^2=hi,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10004255);