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 | |
4470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,16)(11,15,20,24)(12,13,21,23)(17,18,19,22)(25,26)(27,35)(28,34)(29,33)(30,31)(32,38)(36,37)(39,40);(9,11)(10,15)(12,19)(13,22)(14,20)(16,24)(17,21)(18,23)(25,27,31,33)(26,29,30,35)(28,36,38,40)(32,37,34,39);(1,2,5,4)(3,6,7,8)(9,12)(10,13)(11,17)(14,21)(15,18)(16,23)(19,20)(22,24)(25,28)(26,32)(27,36)(29,37)(30,34)(31,38)(33,40)(35,39);(3,7)(6,8)(9,13)(10,12)(11,18)(14,23)(15,17)(16,21)(19,24)(20,22)(25,29,31,35)(26,33,30,27)(28,37,38,39)(32,40,34,36);(1,3)(2,6)(4,8)(5,7)(9,12)(10,13)(11,19)(14,21)(15,22)(16,23)(17,20)(18,24)(25,28)(26,34)(27,36)(29,39)(30,32)(31,38)(33,40)(35,37);(1,4,5,2)(3,8,7,6)(9,11)(10,15)(12,17)(13,18)(14,20)(16,24)(19,21)(22,23)(25,30,31,26)(27,35,33,29)(28,34,38,32)(36,39,40,37)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10365143);