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 | |
4537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,33)(28,34)(29,39)(30,31)(32,37)(35,40)(36,38);(1,3)(2,6)(4,8)(5,7)(11,17)(15,21)(18,23)(22,24)(26,31)(28,38)(32,33)(35,39);(1,2)(3,4)(5,8)(6,7)(10,14)(15,21)(16,19)(22,24)(25,27,30,37)(26,32,31,33)(28,35,38,39)(29,36,40,34);(1,4)(2,7)(3,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,33)(28,35)(29,36)(30,37)(31,32)(34,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,30,38)(26,34,31,36)(27,35,37,39)(29,33,40,32);(1,2)(3,6)(4,7)(5,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,30,40)(26,35,31,39)(27,36,37,34)(28,32,38,33)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9625473);