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 | |
4709 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)(4,6)(5,8)(9,10)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,2,4,6)(3,5,8,7)(10,15)(13,16)(17,20)(21,23)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(17,20)(19,24)(28,37)(32,36)(34,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,29,30,26)(27,31,35,33)(28,36,37,32)(34,38,40,39);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40)Polycyclic presentation:
2^9|b^2=gh,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664747);