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 | |
4632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,15)(12,21)(13,23)(14,16)(17,18)(19,24)(20,22)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,2,4,6)(3,5,7,8)(9,11,14,18)(10,15,16,17)(12,19,23,22)(13,20,21,24)(25,26,29,32)(27,30,35,33)(28,31,37,36)(34,38,40,39);(1,3)(2,5)(4,7)(6,8)(10,16)(11,18)(13,21)(19,22)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(9,11)(10,17)(12,22)(13,20)(14,18)(15,16)(19,23)(21,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,13)(11,19)(14,23)(15,20)(16,21)(17,24)(18,22)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,3)(2,5)(4,7)(6,8)(9,13)(10,12)(11,20)(14,21)(15,19)(16,23)(17,22)(18,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|b^2=h,c^2=g,f^2=g,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, 8840170);