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 | #36022 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,19,23)(12,17,21,24)(13,18,20,22)(25,26)(27,35)(28,38)(29,34)(30,40)(31,33)(32,37)(36,39);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,13)(14,19)(16,23)(17,18)(20,21)(22,24)(25,27,31,37)(26,32,33,35)(28,39,40,34)(29,30,36,38);(2,5)(3,7)(10,16)(11,19)(13,20)(17,24)(26,33)(29,36)(30,38)(32,35);(2,5)(6,8)(9,12,14,21)(10,17,16,24)(11,20,19,13)(15,22,23,18)(25,28)(26,30)(27,34)(29,32)(31,40)(33,38)(35,36)(37,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,20)(10,18,16,22)(11,12,19,21)(15,17,23,24)(25,29,31,36)(26,34,33,39)(27,30,37,38)(28,35,40,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,30)(26,28)(27,36)(29,37)(31,38)(32,39)(33,40)(34,35)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948174);