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 | |
4695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(18,23)(19,24)(25,26)(27,32)(28,34)(29,35)(30,40)(31,33)(36,37)(38,39);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(25,27,31,30)(26,32,33,40)(28,36,35,38)(29,39,34,37);(1,3,5,6)(2,4,7,8)(26,33)(28,35)(32,40)(36,38);(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,28,31,35)(26,34,33,29)(27,36,30,38)(32,37,40,39);(1,2)(3,4)(5,7)(6,8)(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,29,31,34)(26,35,33,28)(27,37,30,39)(32,36,40,38);(1,4)(2,6)(3,7)(5,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,21)(17,24)(25,30,31,27)(26,32,33,40)(28,38,35,36)(29,39,34,37)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9591551);