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 | |
4401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(10,15)(11,17)(16,23)(19,24)(25,26)(27,28)(29,37)(30,32)(31,39)(33,35)(34,38)(36,40);(2,5)(6,7)(9,10,14,15)(11,18,17,12)(13,16,22,23)(19,21,24,20)(25,27)(26,28)(29,34)(30,35)(31,36)(32,33)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(25,26,30,32)(27,33,35,28)(29,31,39,37)(34,38,40,36);(1,3)(2,6)(4,8)(5,7)(9,11,14,17)(10,12,15,18)(13,19,22,24)(16,20,23,21)(25,26)(27,33)(28,35)(29,31)(30,32)(34,38)(36,40)(37,39);(9,12)(10,11)(13,21)(14,18)(15,17)(16,24)(19,23)(20,22)(25,28,30,33)(26,27,32,35)(29,38,39,36)(31,40,37,34);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,29)(26,31)(27,34)(28,36)(30,39)(32,37)(33,38)(35,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154485);