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 | |
4447 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)(10,15)(11,18)(13,21)(20,23)(25,26,30,31)(27,28,35,32)(29,33,38,39)(34,36,37,40);(1,3)(2,7)(4,5)(6,8)(9,10,14,15)(11,12,18,16)(13,17,21,22)(19,20,24,23)(26,31)(28,32)(33,39)(36,40);(1,2,6,4)(3,7,8,5)(9,11,14,18)(10,16,15,12)(13,19,21,24)(17,23,22,20)(25,27)(26,32)(28,31)(29,34)(30,35)(33,40)(36,39)(37,38);(2,4)(3,8)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,24)(22,23)(25,28,30,32)(26,27,31,35)(29,36,38,40)(33,34,39,37);(1,4,6,2)(3,7,8,5)(9,12)(10,11)(13,20)(14,16)(15,18)(17,19)(21,23)(22,24)(25,27)(26,28)(29,37)(30,35)(31,32)(33,40)(34,38)(36,39);(1,5,6,7)(2,3,4,8)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=gi,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, 10227625);