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 | |
4259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,7,5)(9,10)(11,20)(12,13)(14,24)(15,17)(16,21)(18,23)(19,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(2,4)(5,8)(9,11,15,21)(10,16,17,20)(12,22,23,14)(13,24,18,19)(26,30)(31,33)(32,37)(38,39);(2,4)(3,7)(10,17)(11,21)(12,23)(19,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,22)(13,17)(14,21)(15,23)(16,19)(20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,4,6,2)(3,8,7,5)(9,13,15,18)(10,12,17,23)(11,19,21,24)(14,20,22,16);(1,5,6,8)(2,7,4,3)(9,14,15,22)(10,19,17,24)(11,12,21,23)(13,20,18,16)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150873);