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 | C2×#550 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4223 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)(3,7)(4,8)(5,6)(9,10,15,17)(11,18,14,16)(12,13,22,23)(19,20,21,24)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(2,6)(3,8)(9,11,15,14)(10,16,17,18)(12,21,22,19)(13,20,23,24)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(1,3)(2,4)(5,8)(6,7)(10,17)(12,22)(16,18)(19,21)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,13)(11,19)(14,21)(15,22)(16,24)(17,23)(18,20)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,12)(11,20)(14,24)(15,23)(16,21)(17,22)(18,19)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38);(9,14,15,11)(10,16,17,18)(12,19,22,21)(13,20,23,24)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,e^2=gh,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, 8655100);