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 | |
4773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,5,7,4)(9,10,15,13)(11,18,12,16)(14,21,22,17)(19,23,20,24)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,38)(39,40);(1,3)(2,5)(4,8)(6,7)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(1,4,6,5)(2,7,8,3)(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,28,31,29)(26,32,34,27)(30,36,39,37)(33,38,40,35);(1,4,6,5)(2,3,8,7)(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24)(25,27)(26,28)(29,34)(30,38)(31,32)(33,37)(35,39)(36,40);(1,5,6,4)(2,3,8,7)(9,13,15,10)(11,18,12,16)(14,21,22,17)(19,24,20,23)(25,29,31,28)(26,32,34,27)(30,37,39,36)(33,38,40,35);(1,3)(2,4)(5,8)(6,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,38)(34,40)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=h,e^2=gh,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, 10316913);