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 | |
4588 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)(25,26,30,33)(27,29,35,31)(28,32,36,39)(34,37,40,38);(2,5)(6,7)(10,14)(15,21)(16,19)(22,23)(28,36)(32,39)(34,40)(37,38);(1,2)(3,6)(4,5)(7,8)(25,27,30,35)(26,31,33,29)(28,34,36,40)(32,38,39,37);(1,3,4,8)(2,6,5,7)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,27)(26,29)(28,34)(30,35)(31,33)(32,37)(36,40)(38,39);(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,28,30,36)(26,32,33,39)(27,34,35,40)(29,37,31,38);(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,29)(26,27)(28,37)(30,31)(32,34)(33,35)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,e^2=h,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, 9818144);