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 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(25,26,32,36)(27,37,40,33)(28,31,30,34)(29,39,38,35);(3,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,27,32,40)(26,33,36,37)(28,38,30,29)(31,39,34,35);(1,2)(3,6)(4,5)(7,8)(11,17)(15,19)(18,23)(22,24)(25,28,32,30)(26,34,36,31)(27,38,40,29)(33,35,37,39);(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29)(26,35)(27,28)(30,40)(31,33)(32,38)(34,37)(36,39);(1,3)(2,6)(4,7)(5,8)(9,13,14,10)(11,19,17,15)(12,16,20,21)(18,22,23,24)(25,30,32,28)(26,34,36,31)(27,38,40,29)(33,39,37,35);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,31,32,34)(26,28,36,30)(27,39,40,35)(29,33,38,37)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8806751);