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 | |
4382 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)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,24,20,22)(25,26)(27,35)(28,34)(29,32)(30,36)(31,33)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,24,21,22)(25,27,31,29)(26,32,33,35)(28,37,36,39)(30,38,34,40);(10,16)(11,12)(18,21)(19,20)(26,33)(28,36)(32,35)(37,39);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,22,21,24)(25,28,31,36)(26,34,33,30)(27,37,29,39)(32,40,35,38);(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,24,17,22)(25,29,31,27)(26,35,33,32)(28,39,36,37)(30,40,34,38);(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,20,21,19)(25,30,31,34)(26,36,33,28)(27,38,29,40)(32,39,35,37)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7827361);