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 | |
4287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(11,18)(13,16)(19,24)(21,23)(26,32)(27,35)(28,29)(38,39);(1,2)(3,7)(4,5)(6,8)(9,10,15,14)(11,16,18,13)(12,20,22,17)(19,21,24,23)(25,26,31,32)(27,30,35,40)(28,34,29,33)(36,39,37,38);(2,5)(6,7)(9,11,15,18)(10,16,14,13)(12,19,22,24)(17,23,20,21)(25,27)(26,30)(28,36)(29,37)(31,35)(32,40)(33,38)(34,39);(9,12,15,22)(10,17,14,20)(11,19,18,24)(13,21,16,23)(25,28,31,29)(26,33,32,34)(27,36,35,37)(30,38,40,39);(1,3,4,8)(2,6,5,7)(9,13)(10,11)(12,21)(14,18)(15,16)(17,19)(20,24)(22,23)(25,29,31,28)(26,34,32,33)(27,37,35,36)(30,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,14,15,10)(11,13,18,16)(12,20,22,17)(19,21,24,23)(25,30)(26,35)(27,32)(28,38)(29,39)(31,40)(33,37)(34,36)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8122661);