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 | |
4304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(25,26,30,31)(27,33,35,28)(29,37,38,32)(34,36,40,39);(2,6)(3,7)(9,10,14,15)(11,16,19,24)(12,17,22,21)(13,18,23,20)(26,31)(28,33)(32,37)(36,39);(1,2,5,6)(3,8,7,4)(10,15)(11,19)(12,22)(13,23)(25,26)(27,28)(29,32)(30,31)(33,35)(34,36)(37,38)(39,40);(2,6)(3,7)(9,11)(10,16)(12,18)(13,21)(14,19)(15,24)(17,23)(20,22)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(1,3,5,7)(2,4,6,8)(9,12)(10,17)(11,20)(13,16)(14,22)(15,21)(18,19)(23,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,4)(2,7)(3,6)(5,8)(9,13)(10,18)(11,21)(12,16)(14,23)(15,20)(17,19)(22,24)(25,29,30,38)(26,32,31,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8884613);