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 | |
4519 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22);(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(10,11)(13,21)(15,17)(19,24)(26,30)(31,33)(32,36)(38,39);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,3,5,4)(2,7,6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7662019);