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 | |
4644 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)(11,17)(12,13)(14,22)(15,16)(18,23)(19,20)(21,24)(25,26)(27,32)(28,30)(29,36)(31,34)(33,37)(35,38)(39,40);(3,7)(6,8)(9,11,15,13)(10,12,16,17)(14,19,23,21)(18,20,22,24)(27,34)(31,32)(35,40)(38,39);(1,2)(3,6)(4,5)(7,8)(10,16)(12,17)(18,22)(20,24)(25,27,30,34)(26,31,28,32)(29,35,37,40)(33,38,36,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,20)(15,17)(18,19)(21,22)(23,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,17,16,12)(14,21,23,19)(18,24,22,20)(25,28,30,26)(27,32,34,31)(29,36,37,33)(35,39,40,38);(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,22)(17,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|b^2=g,c^2=i,e^2=gi,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, 8072412);