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 | |
4375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,15,18)(11,19,13,16)(12,14,17,24)(20,23,22,21)(25,26)(27,28)(29,32)(30,37)(31,33)(34,39)(35,40)(36,38);(1,2)(3,4)(5,7)(6,8)(9,11,15,13)(10,16,18,19)(12,20,17,22)(14,21,24,23)(25,27,32,31)(26,33,29,28)(30,35,39,38)(34,40,37,36);(1,2)(3,4)(5,7)(6,8)(9,12,15,17)(10,14,18,24)(11,20,13,22)(16,21,19,23)(25,28)(26,31)(27,29)(30,36)(32,33)(34,38)(35,37)(39,40);(1,3,5,8)(2,6,7,4)(9,12)(10,14)(11,20)(13,22)(15,17)(16,21)(18,24)(19,23)(25,29,32,26)(27,33,31,28)(30,37,39,34)(35,40,38,36);(1,2)(3,6)(4,8)(5,7)(9,13,15,11)(10,16,18,19)(12,22,17,20)(14,21,24,23)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40);(1,4,5,6)(2,3,7,8)(9,14)(10,17)(11,21)(12,18)(13,23)(15,24)(16,22)(19,20)(25,31,32,27)(26,33,29,28)(30,38,39,35)(34,40,37,36)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8646565);