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 | |
4409 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10)(11,20)(12,14)(13,18)(15,17)(16,23)(19,24)(21,22)(25,26,30,33)(27,31,36,39)(28,29,37,32)(34,35,40,38);(1,3,5,7)(2,6,8,4)(9,11,15,23)(10,16,17,20)(12,18,19,21)(13,14,22,24)(25,27,30,36)(26,31,33,39)(28,34,37,40)(29,35,32,38);(1,2)(3,4)(5,8)(6,7)(10,17)(12,19)(16,20)(18,21)(25,28,30,37)(26,32,33,29)(27,34,36,40)(31,38,39,35);(1,4)(2,7)(3,8)(5,6)(9,12,15,19)(10,14,17,24)(11,21,23,18)(13,20,22,16)(25,28)(26,29)(27,34)(30,37)(31,35)(32,33)(36,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,13)(10,18)(11,14)(12,20)(15,22)(16,19)(17,21)(23,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,38)(33,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,14)(10,19)(11,22)(12,17)(13,23)(15,24)(16,21)(18,20)(25,29)(26,28)(27,35)(30,32)(31,34)(33,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=ghi,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9846801);