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 | |
4499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,22)(13,23)(14,17)(15,19)(18,20)(21,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,39)(38,40);(1,3,5,8)(2,6,4,7)(9,11,15,21)(10,16,19,24)(12,20,14,13)(17,23,22,18)(25,27,31,39)(26,32,36,37)(28,35,34,38)(29,30,33,40);(1,2,5,4)(3,6,8,7)(9,12,15,14)(10,17,19,22)(11,20,21,13)(16,23,24,18)(25,28,31,34)(26,33,36,29)(27,38,39,35)(30,37,40,32);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,12)(14,21)(15,20)(16,17)(19,23)(22,24)(25,29,31,33)(26,34,36,28)(27,30,39,40)(32,38,37,35);(9,14,15,12)(10,17,19,22)(11,13,21,20)(16,23,24,18)(25,26)(27,32)(28,33)(29,34)(30,38)(31,36)(35,40)(37,39);(9,10)(11,16)(12,17)(13,18)(14,22)(15,19)(20,23)(21,24)(25,30)(26,35)(27,33)(28,37)(29,39)(31,40)(32,34)(36,38)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561439);