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:
(2,6)(7,8)(9,10)(11,17)(12,16)(13,18)(14,19)(15,22)(20,21)(23,24)(25,26,32,31)(27,36,34,29)(28,33,40,39)(30,35,38,37);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)(25,27)(26,29)(28,35)(30,33)(31,36)(32,34)(37,40)(38,39);(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)(25,28)(26,33)(27,37)(29,38)(30,36)(31,39)(32,40)(34,35);(9,12,14,18)(10,16,19,13)(11,20,22,24)(15,23,17,21)(25,29,32,36)(26,34,31,27)(28,38,40,30)(33,35,39,37);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,20)(13,23)(14,22)(15,19)(16,21)(18,24)(25,30,32,38)(26,35,31,37)(27,33,34,39)(28,36,40,29);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,18,19,12)(11,21,22,23)(15,24,17,20)(25,31,32,26)(27,36,34,29)(28,39,40,33)(30,35,38,37)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7565339);