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 | |
4415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,19)(12,21)(13,18)(14,15)(16,20)(17,23)(22,24)(25,26,31,34)(27,32,35,30)(28,29,33,39)(36,37,40,38);(3,7)(6,8)(10,15)(11,20)(17,21)(18,22)(25,27,31,35)(26,32,34,30)(28,36,33,40)(29,37,39,38);(1,2)(3,6)(4,5)(7,8)(10,15)(13,24)(16,19)(17,21)(27,35)(30,32)(36,40)(37,38);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,21)(14,20)(15,19)(17,24)(18,23)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,23)(10,17,15,21)(11,18,20,22)(13,19,24,16)(25,29)(26,33)(27,37)(28,34)(30,36)(31,39)(32,40)(35,38);(9,13)(10,18)(11,21)(12,19)(14,24)(15,22)(16,23)(17,20)(25,30)(26,27)(28,38)(29,36)(31,32)(33,37)(34,35)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8191727);