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:
(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,23,24,18)(25,26)(27,32)(28,33)(29,34)(30,35)(31,38)(36,40)(37,39);(1,2,5,6)(3,7,8,4)(10,15)(11,20)(17,21)(18,23)(25,27,30,38)(26,31,35,32)(28,39,34,36)(29,37,33,40);(1,2)(3,7)(4,8)(5,6)(10,15)(11,20)(12,22)(13,24)(25,27,30,38)(26,32,35,31)(28,39,34,36)(29,40,33,37);(2,6)(3,8)(9,11,14,20)(10,16,15,19)(12,23,22,18)(13,21,24,17)(25,27)(26,31)(28,39)(29,37)(30,38)(32,35)(33,40)(34,36);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,24)(20,23)(25,28,30,34)(26,33,35,29)(27,36,38,39)(31,37,32,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,29,30,33)(26,34,35,28)(27,37,38,40)(31,36,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^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, 8968668);