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 | |
4353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,23)(21,22)(25,26,30,31)(27,33,35,28)(29,37,38,32)(34,36,40,39);(3,7)(6,8)(10,15)(12,22)(16,24)(18,20)(26,31)(28,33)(32,37)(36,39);(1,2)(3,6)(4,5)(7,8)(11,19)(13,23)(16,24)(18,20)(25,26)(27,28)(29,32)(30,31)(33,35)(34,36)(37,38)(39,40);(9,11,14,19)(10,16,15,24)(12,18,22,20)(13,21,23,17)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(1,3,4,7)(2,6,5,8)(9,12,14,22)(10,17,15,21)(11,20,19,18)(13,16,23,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,23)(10,18,15,20)(11,21,19,17)(12,16,22,24)(25,29,30,38)(26,32,31,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^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,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, 8837724);