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 | |
4599 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,26,30,32)(27,31,36,33)(28,34,39,38)(29,37,40,35);(3,7)(5,8)(10,15)(12,19)(17,22)(20,24)(25,27)(26,31)(28,37)(29,38)(30,36)(32,33)(34,40)(35,39);(1,2)(3,5)(4,6)(7,8)(11,16)(12,19)(13,21)(17,22)(26,32)(27,36)(28,39)(29,40);(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,27)(26,33)(28,35)(29,38)(30,36)(31,32)(34,40)(37,39);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,28)(26,34)(27,37)(29,33)(30,39)(31,40)(32,38)(35,36);(1,2)(3,5)(4,6)(7,8)(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,20,19,24)(25,29)(26,35)(27,38)(28,33)(30,40)(31,39)(32,37)(34,36)Polycyclic presentation:
2^9|a^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948301);