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 | |
4455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,17)(11,20,22,16)(12,18,21,14)(13,19,23,24)(25,26)(27,31)(28,39)(29,33)(30,34)(32,38)(35,36)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,22)(10,16,17,20)(12,13,21,23)(14,19,18,24)(25,27)(26,31)(28,37)(29,38)(30,35)(32,33)(34,36)(39,40);(1,3,4,7)(2,5,6,8)(10,17)(14,18)(16,20)(19,24)(27,35)(28,38)(31,36)(32,39);(9,12)(10,18)(11,13)(14,17)(15,21)(16,24)(19,20)(22,23)(25,26)(27,36)(28,39)(29,40)(30,34)(31,35)(32,38)(33,37);(9,13)(10,19)(11,21)(12,22)(14,20)(15,23)(16,18)(17,24)(25,28,30,38)(26,32,34,39)(27,37,35,29)(31,33,36,40);(9,14,15,18)(10,12,17,21)(11,19,22,24)(13,20,23,16)(25,29,30,37)(26,33,34,40)(27,38,35,28)(31,32,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7998941);