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 | |
4603 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)(12,18)(13,14)(15,17)(19,23)(20,22)(21,24)(25,26)(27,34)(28,29)(30,31)(32,37)(33,40)(35,36)(38,39);(3,7)(6,8)(9,11,15,20)(10,16,17,22)(12,19,21,13)(14,18,23,24)(26,31)(28,33)(32,34)(35,38);(1,2)(3,6)(4,5)(7,8)(10,17)(11,20)(12,21)(14,23)(25,27,30,37)(26,32,31,34)(28,35,33,38)(29,39,40,36);(9,12)(10,18)(11,19)(13,20)(14,22)(15,21)(16,23)(17,24)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,35)(39,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,19)(10,14,17,23)(11,21,20,12)(16,24,22,18)(25,28,30,33)(26,29,31,40)(27,35,37,38)(32,39,34,36);(1,2)(3,6)(4,5)(7,8)(9,14)(10,19)(11,18)(12,22)(13,17)(15,23)(16,21)(20,24)(25,29)(26,33)(27,36)(28,31)(30,40)(32,35)(34,38)(37,39)Polycyclic presentation:
2^9|b^2=i,c^2=g,e^2=gi,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, 8824921);