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 | |
4682 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)(11,12)(13,21)(14,15)(16,22)(17,19)(18,24)(20,23)(25,26)(27,34)(28,29)(30,31)(32,35)(33,38)(36,40)(37,39);(2,5)(3,7)(11,17)(12,19)(18,23)(20,24)(26,31)(28,33)(32,34)(36,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,21)(18,23)(27,35)(32,34)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,24)(22,23)(25,27)(26,32)(28,36)(29,37)(30,35)(31,34)(33,39)(38,40);(9,12)(10,11)(13,20)(14,19)(15,17)(16,18)(21,23)(22,24)(25,28,30,33)(26,29,31,38)(27,36,35,39)(32,37,34,40);(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,23)(19,24)(25,29)(26,33)(27,37)(28,31)(30,38)(32,39)(34,36)(35,40)Polycyclic presentation:
2^9|e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000590);