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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4419 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)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26,30,35)(27,31,37,39)(28,33,38,32)(29,36,40,34);(1,2)(3,6)(4,7)(5,8)(10,15)(12,18)(17,21)(20,24)(25,27,30,37)(26,31,35,39)(28,34,38,36)(29,32,40,33);(1,3)(2,5)(4,8)(6,7)(9,11,14,16)(10,12,15,18)(13,22,23,19)(17,24,21,20)(25,28,30,38)(26,32,35,33)(27,36,37,34)(29,39,40,31);(3,8)(5,6)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,28)(26,33)(27,34)(29,31)(30,38)(32,35)(36,37)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,15)(13,20)(14,18)(17,22)(19,21)(23,24)(25,27)(26,31)(28,36)(29,32)(30,37)(33,40)(34,38)(35,39);(1,2)(3,6)(4,7)(5,8)(9,13)(10,17)(11,19)(12,20)(14,23)(15,21)(16,22)(18,24)(25,29,30,40)(26,34,35,36)(27,32,37,33)(28,31,38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8811681);