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 |
| prime radical | #39428 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4434 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)(11,14)(12,13)(15,16)(17,21)(18,24)(19,22)(20,23)(25,26,30,31)(27,33,36,32)(28,34,37,40)(29,38,39,35);(1,2,5,6)(3,7,4,8)(9,11,15,18)(10,14,16,24)(12,19,21,23)(13,22,17,20)(26,31)(28,37)(32,33)(35,38);(1,3,5,4)(2,7,6,8)(10,16)(11,18)(12,21)(20,22)(25,27,30,36)(26,32,31,33)(28,35,37,38)(29,34,39,40);(9,12)(10,17)(11,19)(13,16)(14,20)(15,21)(18,23)(22,24)(25,27)(26,33)(28,35)(29,34)(30,36)(31,32)(37,38)(39,40);(1,4,5,3)(2,8,6,7)(9,13,15,17)(10,12,16,21)(11,20,18,22)(14,23,24,19)(25,28)(26,34)(27,35)(29,33)(30,37)(31,40)(32,39)(36,38);(9,14)(10,18)(11,16)(12,20)(13,19)(15,24)(17,23)(21,22)(25,29)(26,35)(27,34)(28,33)(30,39)(31,38)(32,37)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7948350);