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:
(9,10)(11,19)(12,17)(13,23)(14,15)(16,20)(18,24)(21,22)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,38)(37,40);(1,2,5,6)(3,7,8,4)(10,15)(11,20)(17,21)(18,23)(25,27,29,35)(26,30,33,31)(28,36,37,34)(32,39,40,38);(1,2)(3,7)(4,8)(5,6)(10,15)(11,20)(12,22)(13,24)(25,27,29,35)(26,31,33,30)(28,36,37,34)(32,38,40,39);(2,6)(3,8)(9,11)(10,16)(12,23)(13,21)(14,20)(15,19)(17,24)(18,22)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,22)(10,17,15,21)(11,18,20,23)(13,19,24,16)(25,28,29,37)(26,32,33,40)(27,34,35,36)(30,38,31,39);(1,4)(2,3)(5,7)(6,8)(9,13)(10,18)(11,21)(12,19)(14,24)(15,23)(16,22)(17,20)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,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, 8968332);