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 | |
4479 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)(11,17)(12,16)(13,22)(14,18)(15,21)(19,24)(20,23)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(26,28)(31,33)(32,36)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,22)(11,19,21,20)(15,23,17,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,3)(2,6)(4,8)(5,7)(9,13,14,12)(10,16,18,22)(11,20,21,19)(15,23,17,24)(25,28,30,26)(27,33,35,31)(29,32,38,36)(34,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,17)(12,19)(13,20)(14,21)(15,18)(16,24)(22,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,22)(14,18)(17,21)(19,23)(20,24)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|c^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8566613);