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 | #48685 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4497 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,14)(13,20)(15,18)(17,22)(19,23)(21,24)(25,26)(27,33)(28,30)(29,36)(31,35)(32,37)(34,38)(39,40);(3,7)(6,8)(11,18)(15,16)(19,21)(23,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,23)(20,24)(21,22)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(9,12,14,10)(11,16,18,15)(13,17,22,20)(19,23,21,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,15)(13,21)(14,18)(17,23)(19,22)(20,24)(25,27)(26,33)(28,31)(29,34)(30,35)(32,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9143672);