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 | #41122 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,17,19,12)(13,21,23,16)(18,20,22,24)(25,26,31,33)(27,32,36,30)(28,34,37,40)(29,35,38,39);(3,7)(6,8)(10,15)(12,17)(16,21)(20,24)(25,27)(26,32)(28,29)(30,33)(31,36)(34,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(12,17)(16,21)(20,24)(26,33)(27,36)(29,38)(34,40);(9,11,14,19)(10,12,15,17)(13,22,23,18)(16,24,21,20)(25,28)(26,34)(27,29)(30,39)(31,37)(32,35)(33,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,22)(23,24)(25,29)(26,35)(27,37)(28,36)(30,34)(31,38)(32,40)(33,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,23)(10,16,15,21)(11,18,19,22)(12,20,17,24)(25,30)(26,36)(27,33)(28,35)(29,34)(31,32)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=gi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8972931);