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 | |
4686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,16)(11,17,20,15)(12,18,22,24)(13,19,23,21)(26,28)(31,35)(32,36)(39,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,19)(13,18)(14,20)(16,17)(21,22)(23,24)(25,26,30,28)(27,31,34,35)(29,32,38,36)(33,39,37,40);(10,16)(13,23)(15,17)(18,24)(26,28)(31,35)(32,36)(39,40);(9,11)(10,17)(12,21)(13,18)(14,20)(15,16)(19,22)(23,24)(25,27)(26,31)(28,35)(29,37)(30,34)(32,40)(33,38)(36,39);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,18,16,24)(11,19,20,21)(13,17,23,15)(25,28,30,26)(27,31,34,35)(29,36,38,32)(33,39,37,40);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,19,16,21)(11,18,20,24)(12,17,22,15)(25,29,30,38)(26,32,28,36)(27,33,34,37)(31,39,35,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7976942);