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 | |
4447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(12,20)(16,21)(18,24)(22,23)(25,26,30,31)(27,34,29,32)(28,33,37,38)(35,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(26,31)(32,34)(33,38)(39,40);(10,14)(15,17)(16,21)(22,23)(25,27,30,29)(26,32,31,34)(28,36,37,35)(33,40,38,39);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,30,37)(26,33,31,38)(27,35,29,36)(32,39,34,40);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,27)(26,34,31,32)(28,36,37,35)(33,40,38,39)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551115);