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:
(1,2,5,8)(3,6,4,7)(11,18)(16,22)(19,24)(20,23)(25,26)(27,32)(28,37)(29,40)(30,33)(31,34)(35,36)(38,39);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,23)(20,24)(25,27,31,38)(26,32,34,39)(28,33,40,35)(29,36,37,30);(10,15)(12,21)(16,22)(19,24)(25,28,31,40)(26,29,34,37)(27,35,38,33)(30,39,36,32);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24)(25,28)(26,29)(27,33)(30,39)(31,40)(32,36)(34,37)(35,38);(1,4,5,3)(2,7,8,6)(9,12,14,21)(10,17,15,13)(11,19,18,24)(16,23,22,20)(25,29)(26,28)(27,36)(30,38)(31,37)(32,33)(34,40)(35,39);(9,13,14,17)(10,12,15,21)(11,20,18,23)(16,19,22,24)(25,30,31,36)(26,33,34,35)(27,37,38,29)(28,39,40,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7895344);