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 | |
4471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,31,33)(27,36,30,32)(28,34,37,39)(29,35,40,38);(2,6)(4,7)(12,19)(16,22)(18,23)(20,24)(25,27,31,30)(26,32,33,36)(28,35,37,38)(29,39,40,34);(1,2)(3,4)(5,6)(7,8)(10,15)(11,17)(16,22)(18,23)(26,33)(28,37)(32,36)(35,38);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28)(26,34)(27,35)(29,32)(30,38)(31,37)(33,39)(36,40);(1,4,5,7)(2,3,6,8)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,29)(26,35)(27,34)(28,32)(30,39)(31,40)(33,38)(36,37);(1,3)(2,4)(5,8)(6,7)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,30,31,27)(26,36,33,32)(28,38,37,35)(29,34,40,39)Polycyclic presentation:
2^9|a^2=hi,b^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,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, 9625785);