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 | |
4187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,20)(13,15)(14,19)(17,21)(18,22)(23,24)(25,26,30,31)(27,34,29,32)(28,33,36,38)(35,40,37,39);(3,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,16)(17,22)(20,23)(21,24)(26,31)(28,36)(32,34)(35,37);(1,2)(3,6)(4,5)(7,8)(10,15)(12,21)(14,16)(18,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,10)(11,14)(12,17)(13,15)(16,19)(18,22)(20,21)(23,24)(25,27,30,29)(26,32,31,34)(28,35,36,37)(33,39,38,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,18)(13,19)(14,15)(17,23)(20,22)(21,24)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,12,13,21)(10,17,15,20)(11,18,19,24)(14,22,16,23)(25,29,30,27)(26,34,31,32)(28,37,36,35)(33,40,38,39)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837717);