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 | |
4526 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)(9,10,14,16)(11,17,12,15)(13,18,22,21)(19,24,20,23)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,3,5,8)(2,6,4,7)(9,11,14,12)(10,15,16,17)(13,20,22,19)(18,24,21,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,2,5,4)(3,6,8,7)(10,16)(11,12)(18,21)(19,20)(26,28)(31,33)(32,36)(38,39);(1,2,5,4)(3,6,8,7)(9,12,14,11)(10,17,16,15)(13,20,22,19)(18,24,21,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(9,10)(11,15)(12,17)(13,21)(14,16)(18,22)(19,24)(20,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,18,16,21)(11,19,12,20)(15,23,17,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,d^2=g,f^2=g,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, 7876873);