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 | |
4334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,11)(12,17,20,15)(13,16,22,18)(19,24,21,23)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40);(3,7)(5,8)(10,11)(15,17)(16,18)(23,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,11)(15,17)(16,18)(23,24)(27,34)(30,31)(35,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,26)(27,30)(28,36)(29,33)(31,34)(32,37)(35,39)(38,40)Polycyclic presentation:
2^9|a^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8634018);