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 | |
4450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(25,26,30,32)(27,31,35,39)(28,33,38,37)(29,36,40,34);(1,2)(3,7)(4,5)(6,8)(10,15)(12,17)(16,22)(20,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,38)(37,40);(2,5)(6,7)(13,21)(16,22)(18,24)(20,23)(26,32)(27,35)(33,37)(34,36);(3,8)(6,7)(9,11,14,19)(10,12,15,17)(13,18,21,24)(16,20,22,23)(25,28)(26,33)(27,36)(29,39)(30,38)(31,40)(32,37)(34,35);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,29,30,40)(26,34,32,36)(27,37,35,33)(28,31,38,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,40)(34,38)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8254585);