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 | |
4406 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)(11,16)(12,18)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26,29,31)(27,30,34,33)(28,36,37,32)(35,39,40,38);(3,7)(6,8)(10,15)(11,18)(17,21)(19,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(13,22)(20,23)(26,31)(27,34)(32,36)(35,40);(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,26)(27,33)(28,36)(29,31)(30,34)(32,37)(35,38)(39,40);(1,3,4,7)(2,6,5,8)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,28,29,37)(26,32,31,36)(27,35,34,40)(30,38,33,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,27)(26,33)(28,35)(29,34)(30,31)(32,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841077);