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 | |
4430 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)(11,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,24)(25,26)(27,32)(28,38)(29,34)(30,35)(31,36)(33,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,15)(10,12,17,19)(13,18,22,23)(16,20,21,24)(25,27)(26,32)(28,29)(30,36)(31,35)(33,40)(34,38)(37,39);(11,15)(12,19)(18,23)(20,24)(25,28,31,39)(26,33,36,38)(27,37,35,29)(30,40,32,34);(9,10)(11,12)(13,21)(14,17)(15,19)(16,22)(18,24)(20,23)(25,29)(26,34)(27,28)(30,33)(31,37)(32,38)(35,39)(36,40);(1,3,4,7)(2,5,6,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,23)(18,21)(22,24)(25,28)(26,33)(27,29)(30,34)(31,39)(32,40)(35,37)(36,38);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,30)(26,35)(27,36)(28,34)(29,38)(31,32)(33,37)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7980338);