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 | #57178 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(6,7)(9,10,16,15)(11,19,14,17)(12,18,13,22)(20,24,21,23)(25,26)(27,35)(28,34)(29,33)(30,31)(32,37)(36,39)(38,40);(1,2,5,4)(3,7,8,6)(9,11,16,14)(10,17,15,19)(12,20,13,21)(18,23,22,24)(25,27)(26,32)(28,36)(29,38)(30,35)(31,37)(33,39)(34,40);(1,3)(2,6)(4,7)(5,8)(9,12,16,13)(10,18,15,22)(11,20,14,21)(17,23,19,24)(25,28,31,29)(26,33,30,34)(27,36,37,38)(32,39,35,40);(2,4)(6,7)(9,13,16,12)(10,18,15,22)(11,21,14,20)(17,23,19,24)(25,29,31,28)(26,34,30,33)(27,36,37,38)(32,39,35,40);(1,3)(2,7)(4,6)(5,8)(9,14,16,11)(10,17,15,19)(12,21,13,20)(18,23,22,24)(25,30,31,26)(27,35,37,32)(28,34,29,33)(36,40,38,39);(1,4,5,2)(3,7,8,6)(9,15,16,10)(11,19,14,17)(12,22,13,18)(20,24,21,23)(25,27)(26,35)(28,36)(29,38)(30,32)(31,37)(33,40)(34,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8570080);