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 | |
4603 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,8,7)(9,10)(11,16)(12,18)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26,31,29)(27,34,28,32)(30,37,38,33)(35,39,36,40);(1,2,4,6)(3,7,8,5)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,23)(22,24)(25,26,31,29)(27,32,28,34)(30,33,38,37)(35,39,36,40);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(17,21)(19,24)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,3,4,8)(2,5,6,7)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,21)(22,24)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=hi,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002369);