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 | |
4384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,21,16)(12,17,22,20)(13,23,24,18)(25,26)(27,32)(28,38)(29,34)(30,35)(31,37)(33,36)(39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(12,22)(16,19)(18,23)(25,27,30,37)(26,31,35,32)(28,29,36,39)(33,34,38,40);(2,5)(6,8)(10,15)(13,24)(16,19)(17,20)(25,27,30,37)(26,32,35,31)(28,29,36,39)(33,40,38,34);(3,7)(6,8)(9,11,14,21)(10,16,15,19)(12,23,22,18)(13,20,24,17)(25,28)(26,33)(27,29)(30,36)(31,34)(32,40)(35,38)(37,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,18)(13,19)(14,22)(15,20)(16,24)(21,23)(25,29)(26,34)(27,36)(28,37)(30,39)(31,38)(32,33)(35,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,24)(10,18,15,23)(11,20,21,17)(12,19,22,16)(25,26)(27,31)(28,33)(29,34)(30,35)(32,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800422);