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 | |
4413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,5,8,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,23)(22,24)(25,26)(27,32)(28,30)(29,33)(31,35)(34,40)(36,38)(37,39);(1,3)(2,7)(4,5)(6,8)(9,11,14,16)(10,15,18,12)(13,19,22,23)(17,21,24,20)(26,28)(31,32)(33,36)(39,40);(1,2,6,4)(3,7,8,5)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,27)(26,31)(28,32)(29,34)(30,35)(33,39)(36,40)(37,38);(2,4)(3,8)(9,11)(10,15)(12,18)(13,19)(14,16)(17,21)(20,24)(22,23)(25,28,30,26)(27,32,35,31)(29,36,38,33)(34,40,37,39);(1,4,6,2)(3,7,8,5)(9,12,14,15)(10,16,18,11)(13,21,22,20)(17,19,24,23)(25,27)(26,32)(28,31)(29,37)(30,35)(33,39)(34,38)(36,40);(1,5,6,7)(2,3,4,8)(9,13,14,22)(10,17,18,24)(11,19,16,23)(12,20,15,21)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=hi,e^2=gh,f^2=gh,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, 10227626);