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 | |
4449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,12)(13,21,22,16)(18,20,24,23)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,38)(37,40);(1,2,5,6)(3,7,8,4)(9,11,14,19)(10,12,15,17)(13,18,22,24)(16,20,21,23)(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)(12,17)(13,22)(18,24)(25,27,29,35)(26,31,33,30)(28,36,37,34)(32,38,40,39);(2,6)(3,8)(9,10)(11,17)(12,19)(13,16)(14,15)(18,23)(20,24)(21,22)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,17)(10,11,15,19)(13,20,22,23)(16,18,21,24)(25,28,29,37)(26,32,33,40)(27,34,35,36)(30,38,31,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,22)(10,16,15,21)(11,18,19,24)(12,20,17,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,e^2=g,f^2=h,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, 8968777);