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 | #47937 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4242 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)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,26,30,34)(27,31,38,35)(28,32,29,33)(36,39,37,40);(1,2,4,6)(3,7,8,5)(10,15)(14,21)(16,19)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(3,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,40)(34,35);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,35,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,17)(13,20)(14,22)(15,19)(18,24)(21,23)(25,29,30,28)(26,33,34,32)(27,37,38,36)(31,40,35,39)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,e^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=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9397580);