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 | |
4515 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)(7,8)(9,10)(11,15)(12,20)(13,17)(14,18)(16,21)(19,22)(23,24)(25,26)(27,32)(28,30)(29,38)(31,33)(34,40)(35,39)(36,37);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,31,29)(26,32,33,38)(28,37,34,35)(30,36,40,39);(11,18)(14,15)(19,24)(22,23)(26,33)(27,29)(28,34)(36,39);(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,21)(22,24)(25,28,31,34)(26,30,33,40)(27,35,29,37)(32,39,38,36);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,29,31,27)(26,32,33,38)(28,37,34,35)(30,39,40,36);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,17,20)(11,19,18,24)(14,22,15,23)(25,30)(26,34)(27,36)(28,33)(29,39)(31,40)(32,35)(37,38)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621774);