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 | |
4377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,30,33)(27,34,32,29)(28,31,39,37)(35,38,36,40);(1,2,4,5)(3,6,7,8)(11,17)(15,21)(18,23)(22,24)(25,27)(26,29)(28,36)(30,32)(31,38)(33,34)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(10,14)(15,21)(16,19)(22,24)(25,26,30,33)(27,29,32,34)(28,31,39,37)(35,40,36,38);(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,26)(27,34)(28,37)(29,32)(30,33)(31,39)(35,40)(36,38);(1,2)(3,8)(4,5)(6,7)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28)(26,31)(27,35)(29,40)(30,39)(32,36)(33,37)(34,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,30,34)(26,32,33,27)(28,38,39,40)(31,35,37,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10219604);