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 | |
4437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,18)(12,17)(13,22)(14,15)(16,19)(20,24)(21,23)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,2,6,5)(3,8,7,4)(10,15)(12,22)(16,18)(20,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(10,15)(11,19)(13,17)(20,23)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(1,2,6,5)(3,4,7,8)(9,11,14,19)(10,16,15,18)(12,23,22,20)(13,21,17,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,4,6,8)(2,7,5,3)(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,24)(19,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,5,6,2)(3,8,7,4)(9,13,14,17)(10,12,15,22)(11,21,19,24)(16,23,18,20)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=gh,d^2=g,e^2=ghi,f^2=h,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, 8826585);