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 | |
4452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,31,33)(27,32,30,40)(28,34,37,29)(35,36,38,39);(1,2)(3,8)(4,6)(5,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27,31,30)(26,32,33,40)(28,35,37,38)(29,39,34,36);(1,3,5,6)(2,4,7,8)(10,15)(11,18)(16,21)(19,24)(26,33)(28,37)(32,40)(35,38);(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,28)(26,34)(27,35)(29,33)(30,38)(31,37)(32,36)(39,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29,31,34)(26,28,33,37)(27,36,30,39)(32,38,40,35);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,30,31,27)(26,32,33,40)(28,38,37,35)(29,39,34,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9591216);