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 | |
4748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(13,21)(16,22)(18,23)(19,24)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,3)(2,5)(4,7)(6,8)(10,15)(11,17)(16,22)(18,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gi,f^2=g,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, 8066562);