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 | |
4702 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)(26,31)(27,28)(33,38)(34,36);(1,2)(3,7)(4,5)(6,8)(10,15)(12,17)(16,22)(20,23)(25,26,30,31)(27,32,28,35)(29,33,37,38)(34,39,36,40);(2,5)(6,7)(13,21)(16,22)(18,24)(20,23)(25,27,30,28)(26,32,31,35)(29,36,37,34)(33,40,38,39);(3,8)(6,7)(9,11,14,19)(10,12,15,17)(13,18,21,24)(16,20,22,23)(25,28,30,27)(26,32,31,35)(29,36,37,34)(33,39,38,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,26)(27,32)(28,35)(29,33)(30,31)(34,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,29,30,37)(26,33,31,38)(27,34,28,36)(32,39,35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=g,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, 8254074);