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 | #2456 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,21,24)(12,18,19,13)(14,20,23,22)(25,26,31,35)(27,29,37,32)(28,38,36,33)(30,34,40,39);(2,5)(6,8)(9,11,15,21)(10,16,17,24)(12,20,19,22)(13,14,18,23)(25,27,31,37)(26,32,35,29)(28,30,36,40)(33,34,38,39);(1,2,4,5)(3,6,7,8)(10,17)(11,21)(13,18)(20,22)(25,28,31,36)(26,33,35,38)(27,30,37,40)(29,39,32,34);(3,7)(6,8)(9,12,15,19)(10,18,17,13)(11,22,21,20)(14,24,23,16)(25,27)(26,32)(28,30)(29,35)(31,37)(33,34)(36,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,13,15,18)(10,19,17,12)(11,23,21,14)(16,20,24,22)(25,29,31,32)(26,27,35,37)(28,34,36,39)(30,33,40,38);(1,3)(2,6)(4,7)(5,8)(9,14,15,23)(10,20,17,22)(11,18,21,13)(12,16,19,24)(25,30)(26,34)(27,36)(28,37)(29,33)(31,40)(32,38)(35,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8509247);