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 | |
4482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,23,21)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,40)(38,39);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,23)(22,24)(25,27,30,37)(26,31,34,35)(28,29,36,39)(32,38,40,33);(9,11,14,19)(10,15,17,12)(13,18,22,23)(16,21,24,20)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,40)(38,39);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28,30,36)(26,32,34,40)(27,29,37,39)(31,38,35,33);(1,2,5,4)(3,6,8,7)(9,12)(10,11)(13,21)(14,15)(16,23)(17,19)(18,24)(20,22)(25,26)(27,31)(28,32)(29,38)(30,34)(33,39)(35,37)(36,40);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,16,17,24)(11,18,19,23)(12,20,15,21)(25,29)(26,33)(27,36)(28,37)(30,39)(31,32)(34,38)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7796166);