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 | |
4577 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)(11,16)(12,18)(13,19)(14,20)(15,17)(21,24)(22,23)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,2,4,6)(3,5,8,7)(9,11)(10,16)(12,13)(14,17)(15,20)(18,19)(21,22)(23,24);(1,2)(3,5)(4,6)(7,8)(10,17)(11,20)(13,21)(18,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,21)(13,20)(14,19)(15,22)(16,24)(17,23)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(9,13,15,21)(10,19,17,24)(11,12,20,22)(14,23,16,18)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,14)(10,11)(12,19)(13,23)(15,16)(17,20)(18,21)(22,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10243972);