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 | |
4635 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,18)(16,20)(22,24)(25,26,30,33)(27,34,28,31)(29,37,38,32)(35,39,36,40);(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,17,18)(12,20,21,16)(19,24,23,22)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(2,5)(6,7)(11,17)(15,18)(19,23)(22,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(2,5)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,29,30,38)(26,32,33,37)(27,35,28,36)(31,39,34,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10228128);