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 | |
4824 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,26,31,33)(27,30,28,32)(29,34,38,40)(35,37,36,39);(2,5)(6,7)(10,14)(15,21)(16,19)(22,23)(25,27,31,28)(26,32,33,30)(29,36,38,35)(34,37,40,39);(1,2)(3,6)(4,5)(7,8)(26,33)(27,28)(34,40)(35,36);(1,3,4,8)(2,6,5,7)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,28,31,27)(26,30,33,32)(29,36,38,35)(34,37,40,39);(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40);(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,30)(26,27)(28,33)(29,37)(31,32)(34,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817201);