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 | |
4319 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)(3,7,8,5)(9,10,14,17)(11,12,19,15)(13,21,22,16)(18,23,24,20)(26,30)(28,36)(31,33)(34,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,21,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,21)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066145);