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 | |
4311 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,21)(13,22)(14,16)(15,20)(18,19)(23,24)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,23)(22,24)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(1,2)(3,5)(4,6)(7,8)(10,16)(12,22)(15,17)(18,24)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,18)(14,22)(15,19)(16,21)(17,23)(20,24)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(9,11)(10,17)(12,18)(13,23)(14,20)(15,16)(19,21)(22,24)(25,29,31,27)(26,33,28,32)(30,35,38,37)(34,39,36,40);(9,13)(10,12)(11,19)(14,21)(15,18)(16,22)(17,24)(20,23)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=ghi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9817005);