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 | |
4677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(10,14)(11,17)(16,22)(19,20)(25,26,29,30)(27,33,35,31)(28,36,38,32)(34,39,37,40);(1,2,4,6)(3,5,8,7)(9,10,13,14)(11,15,17,18)(12,16,21,22)(19,23,20,24)(26,30)(31,33)(32,36)(39,40);(1,2)(3,5)(4,6)(7,8)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38);(3,8)(5,7)(9,11)(10,15)(12,20)(13,17)(14,18)(16,24)(19,21)(22,23)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,23)(21,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12,13,21)(10,16,14,22)(11,19,17,20)(15,23,18,24)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315285);