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 | |
4454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,7)(13,21)(18,22)(19,24)(20,23)(26,31)(27,29)(28,36)(39,40);(2,6)(3,8)(10,15)(12,16)(18,22)(20,23)(25,26,30,31)(27,32,29,34)(28,37,36,33)(35,40,38,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,27,30,29)(26,32,31,34)(28,35,36,38)(33,39,37,40);(1,3,5,8)(2,7,6,4)(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20)(25,28,30,36)(26,33,31,37)(27,35,29,38)(32,39,34,40);(1,2)(3,7)(4,8)(5,6)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(1,4,5,7)(2,8,6,3)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29,30,27)(26,34,31,32)(28,38,36,35)(33,40,37,39)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017652);