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 | |
4310 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)(3,4)(5,8)(6,7)(9,10)(11,18)(12,17)(13,15)(14,16)(19,24)(20,22)(21,23)(25,26)(27,37)(28,31)(29,34)(30,35)(32,36)(33,40)(38,39);(1,3,5,7)(2,4,8,6)(9,11,14,13)(10,15,16,18)(12,20,21,19)(17,24,23,22)(25,27)(26,33)(28,30)(29,38)(31,39)(32,40)(34,35)(36,37);(1,3)(2,6)(4,8)(5,7)(10,16)(15,18)(17,23)(22,24)(25,28,32,29)(26,31,36,34)(27,30,40,38)(33,39,37,35);(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,29,32,28)(26,34,36,31)(27,38,40,30)(33,35,37,39);(1,3)(2,4)(5,7)(6,8)(9,12,14,21)(10,17,16,23)(11,19,13,20)(15,22,18,24)(25,30,32,38)(26,35,36,39)(27,28,40,29)(31,33,34,37);(1,4)(2,7)(3,8)(5,6)(9,13,14,11)(10,18,16,15)(12,19,21,20)(17,22,23,24)(25,31,32,34)(26,28,36,29)(27,39,40,35)(30,37,38,33)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,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, 9633805);