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 | #50059 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4211 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)(12,16)(13,21)(14,15)(18,22)(19,23)(20,24)(25,26,31,35)(27,29,33,40)(28,34,30,32)(36,39,37,38);(1,2)(3,5)(4,6)(7,8)(10,15)(16,17)(18,21)(23,24)(25,27)(26,29)(28,36)(30,37)(31,33)(32,38)(34,39)(35,40);(1,2,4,6)(3,7,8,5)(9,11,14,12)(10,16,15,17)(13,20,22,19)(18,24,21,23)(25,28,31,30)(26,32,35,34)(27,36,33,37)(29,38,40,39);(3,8)(5,7)(9,12,14,11)(10,17,15,16)(13,20,22,19)(18,24,21,23);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,23)(21,22)(25,29)(26,33)(27,35)(28,38)(30,39)(31,40)(32,37)(34,36);(1,2)(3,5)(4,6)(7,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,30,31,28)(26,34,35,32)(27,37,33,36)(29,39,40,38)Polycyclic presentation:
2^9|a^2=gh,c^2=hi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8850707);