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 | |
4450 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)(11,17)(15,18)(19,23)(22,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,3,5,8)(2,6,4,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27,31,38)(26,32,36,37)(28,30,34,39)(29,35,33,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,31,34)(26,33,36,29)(27,30,38,39)(32,40,37,35);(1,4,5,2)(3,7,8,6)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,29,31,33)(26,34,36,28)(27,35,38,40)(30,32,39,37);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,30)(26,35)(27,34)(28,38)(29,37)(31,39)(32,33)(36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=g,e^2=gh,f^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, 7561178);