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 |
| prime radical | #35696 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,18)(13,19)(14,22)(15,17)(20,24)(21,23)(25,26)(27,29)(28,37)(30,38)(31,33)(32,36)(34,35)(39,40);(1,2)(3,4)(5,7)(6,8)(9,11,15,12)(10,16,17,18)(13,22,23,20)(14,21,24,19)(25,27,31,36)(26,32,33,29)(28,34,38,40)(30,35,37,39);(1,3)(2,6)(4,7)(5,8)(10,17)(11,12)(13,23)(14,24)(25,28,31,38)(26,30,33,37)(27,34,36,40)(29,39,32,35);(3,8)(4,6)(9,12,15,11)(10,18,17,16)(13,20,23,22)(14,19,24,21)(25,28)(26,30)(27,34)(29,39)(31,38)(32,35)(33,37)(36,40);(1,4)(2,3)(5,6)(7,8)(9,13,15,23)(10,19,17,21)(11,20,12,22)(14,16,24,18)(25,29,31,32)(26,27,33,36)(28,35,38,39)(30,40,37,34);(1,2)(3,4)(5,7)(6,8)(9,14)(10,20)(11,21)(12,19)(13,16)(15,24)(17,22)(18,23)(25,30)(26,28)(27,35)(29,40)(31,37)(32,34)(33,38)(36,39)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8877124);