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 | |
4644 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)(9,10,14,16)(11,15,17,24)(12,13,21,23)(18,19,20,22)(26,31)(27,32)(33,37)(34,39);(3,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,17)(16,24)(18,21)(19,23)(25,26,30,31)(27,28,32,35)(29,37,38,33)(34,40,39,36);(1,2)(3,6)(4,5)(7,8)(10,16)(11,17)(12,21)(19,22)(27,32)(28,35)(34,39)(36,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,24)(20,21)(22,23)(25,27,30,32)(26,28,31,35)(29,34,38,39)(33,36,37,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,13)(11,18)(14,21)(15,19)(16,23)(17,20)(22,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,12)(11,19)(14,23)(15,18)(16,21)(17,22)(20,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8668893);