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 | |
4390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(2,3)(6,7)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(28,36)(31,38)(34,40)(37,39);(1,2,5,3)(4,6,8,7)(10,15)(12,22)(16,24)(18,19)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,19)(13,17)(14,21)(15,24)(18,22)(20,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8424752);