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 | #40793 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(11,19)(12,22)(18,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2,5,3)(4,6,8,7)(9,10,15,14)(11,20,19,16)(12,17,22,23)(13,24,21,18)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(2,3)(6,7)(9,11,15,19)(10,16,14,20)(12,21,22,13)(17,24,23,18)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(2,3)(4,8)(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,3,5,2)(4,7,8,6)(9,13,15,21)(10,18,14,24)(11,12,19,22)(16,17,20,23)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,10)(11,20,19,16)(12,23,22,17)(13,24,21,18)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012484);