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 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,21)(12,18)(13,24)(14,20)(15,17)(16,23)(19,22)(25,26,30,33)(27,28,35,31)(29,32,39,40)(34,36,37,38);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,19)(13,18)(14,22)(15,23)(17,21)(20,24)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,39)(38,40);(1,3,5,4)(2,7,6,8)(10,17)(12,20)(16,21)(19,24)(25,27,30,35)(26,31,33,28)(29,34,39,37)(32,38,40,36);(9,12,15,20)(10,18,17,14)(11,19,23,24)(13,21,22,16)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,40)(37,39);(9,13,15,22)(10,19,17,24)(11,18,23,14)(12,21,20,16)(25,28,30,31)(26,27,33,35)(29,38,39,36)(32,37,40,34);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,20,17,12)(11,22,23,13)(16,24,21,19)(25,29)(26,32)(27,34)(28,36)(30,39)(31,38)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=ghi,d^2=hi,e^2=h,f^2=gi,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, 7894415);