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 | |
4598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,40)(38,39);(2,5)(3,7)(12,21)(15,20)(18,24)(22,23)(25,27)(26,32)(28,38)(29,35)(30,39)(31,40)(33,34)(36,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,28,31,35)(26,33,36,30)(27,38,40,29)(32,34,37,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,23)(17,20)(19,24);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,29,31,38)(26,34,36,39)(27,35,40,28)(30,32,33,37);(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,30,31,33)(26,35,36,28)(27,39,40,34)(29,37,38,32)Polycyclic presentation:
2^9|c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8004963);