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 | #48526 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,30,35)(27,31,37,32)(28,33,38,39)(29,36,40,34);(1,2)(3,6)(4,7)(5,8)(10,15)(12,17)(16,21)(20,23)(25,27)(26,31)(28,34)(29,33)(30,37)(32,35)(36,38)(39,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,18)(13,22)(20,23)(25,27,30,37)(26,32,35,31)(28,36,38,34)(29,39,40,33);(3,8)(5,6)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27)(26,31)(28,34)(29,33)(30,37)(32,35)(36,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,28)(26,33)(27,36)(29,32)(30,38)(31,40)(34,37)(35,39);(1,2)(3,6)(4,7)(5,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,29,30,40)(26,34,35,36)(27,33,37,39)(28,32,38,31)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8811649);