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 | |
4543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(13,21)(16,18)(19,24)(22,23)(25,26)(27,34)(28,31)(29,33)(30,36)(32,37)(35,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27)(26,30)(28,35)(29,36)(31,38)(32,39)(33,34)(37,40);(1,3)(2,4)(5,8)(6,7)(10,11)(15,17)(16,18)(22,23)(25,28,29,37)(26,31,33,32)(27,35,36,40)(30,38,34,39);(2,6)(3,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28)(26,32)(27,35)(29,37)(30,39)(31,33)(34,38)(36,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,26)(27,30)(28,31)(29,33)(32,37)(34,36)(35,38)(39,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,27)(26,30)(28,35)(29,36)(31,38)(32,39)(33,34)(37,40)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=i,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8074581);