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 | |
4397 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)(11,18)(12,17)(13,22)(14,15)(16,21)(19,24)(20,23)(25,26)(27,32)(28,38)(29,33)(30,36)(31,35)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(10,15)(12,22)(16,18)(19,23)(25,27,31,37)(26,32,35,39)(28,36,29,34)(30,33,40,38);(2,5)(6,8)(10,15)(13,17)(16,18)(20,24)(25,28,31,29)(26,33,35,38)(27,34,37,36)(30,39,40,32);(3,7)(6,8)(9,11,14,21)(10,16,15,18)(12,23,22,19)(13,20,17,24)(25,26)(27,32)(28,33)(29,38)(30,34)(31,35)(36,40)(37,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,24)(21,23)(25,29,31,28)(26,33,35,38)(27,36,37,34)(30,39,40,32);(1,2)(3,6)(4,5)(7,8)(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,23,18,19)(25,30)(26,34)(27,33)(28,32)(29,39)(31,40)(35,36)(37,38)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=g,e^2=i,f^2=g,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, 8806267);