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 | #55271 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,13,15)(11,14,17,22)(12,16,19,23)(18,21,20,24)(25,26,30,31)(27,34,29,32)(28,33,36,38)(35,40,37,39);(3,7)(5,8)(9,11,13,17)(10,14,15,22)(12,18,19,20)(16,21,23,24)(26,31)(28,36)(32,34)(35,37);(1,2)(3,5)(4,6)(7,8)(10,15)(11,17)(12,19)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,12)(10,16)(11,18)(13,19)(14,21)(15,23)(17,20)(22,24)(25,27,30,29)(26,32,31,34)(28,35,36,37)(33,39,38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,20)(13,17)(15,22)(16,24)(18,19)(21,23)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,23)(20,24)(25,29,30,27)(26,34,31,32)(28,37,36,35)(33,40,38,39)Polycyclic presentation:
2^9|a^2=h,b^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,d^c=dgi,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, 8883542);