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 | #58082 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,17)(11,14,19,15)(12,16,21,24)(18,22,20,23)(25,26,30,31)(27,33,35,28)(29,37,38,32)(34,36,40,39);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,20)(13,19)(15,17)(16,23)(18,21)(22,24)(26,31)(28,33)(32,37)(36,39);(1,3)(2,6)(4,7)(5,8)(9,11,13,19)(10,15,17,14)(12,20,21,18)(16,22,24,23)(25,26)(27,28)(29,32)(30,31)(33,35)(34,36)(37,38)(39,40);(3,8)(4,6)(9,11)(10,14)(12,20)(13,19)(15,17)(16,23)(18,21)(22,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(1,4)(2,3)(5,6)(7,8)(9,12,13,21)(10,16,17,24)(11,18,19,20)(14,22,15,23)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,23)(21,24)(25,29,30,38)(26,32,31,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=gh,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, 8876844);