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 | |
4590 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)(3,4)(5,8)(6,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,26)(27,30)(28,35)(29,31)(32,36)(33,39)(34,37)(38,40);(1,3)(2,4)(5,6)(7,8)(10,12)(11,17)(16,20)(19,23)(25,27,29,34)(26,30,31,37)(28,33,36,40)(32,38,35,39);(1,4)(2,6)(3,8)(5,7)(26,31)(30,37)(32,35)(38,39);(1,2,5,8)(3,7,6,4)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,26)(27,30)(28,32)(29,31)(33,38)(34,37)(35,36)(39,40);(1,4)(2,3)(5,7)(6,8)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,3)(2,7)(4,8)(5,6)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,28)(26,32)(27,33)(29,36)(30,38)(31,35)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9854164);