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 | |
4773 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)(12,21)(13,23)(14,17)(15,18)(19,20)(22,24)(25,26,30,33)(27,34,31,28)(29,37,38,32)(35,36,39,40);(3,7)(6,8)(11,18)(15,16)(19,22)(20,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,18)(10,15,17,16)(12,19,23,22)(13,20,21,24)(25,27,30,31)(26,28,33,34)(29,35,38,39)(32,36,37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,22)(13,20)(14,18)(15,17)(19,23)(21,24)(25,28,30,34)(26,31,33,27)(29,36,38,40)(32,39,37,35);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,19)(14,23)(15,20)(16,24)(17,21)(18,22)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,38)(39,40);(9,13)(10,12)(11,20)(14,21)(15,19)(16,22)(17,23)(18,24)(25,29,30,38)(26,32,33,37)(27,35,31,39)(28,36,34,40)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064007);