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 | #40150 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,17)(16,21)(22,23)(25,26,31,35)(27,34,29,32)(28,33,38,30)(36,40,39,37);(1,2)(3,4)(5,6)(7,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(27,29)(32,34)(36,39)(37,40);(2,6)(4,7)(12,20)(16,21)(18,24)(22,23)(25,27,31,29)(26,32,35,34)(28,36,38,39)(30,40,33,37);(2,6)(3,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28)(26,33)(27,36)(29,39)(30,35)(31,38)(32,37)(34,40);(1,3)(2,7)(4,6)(5,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,27)(26,34,35,32)(28,36,38,39)(30,40,33,37);(1,4,5,7)(2,3,6,8)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30,31,33)(26,28,35,38)(27,37,29,40)(32,39,34,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8306309);