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 | #51333 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4065 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)(11,15)(12,17)(13,21)(14,16)(18,23)(19,22)(20,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,15,16,22)(12,18,20,21)(13,17,23,24)(25,26,30,34)(27,31,38,35)(28,32,29,33)(36,39,37,40);(2,5)(3,7)(10,16)(11,19)(12,20)(13,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(3,7)(6,8)(9,12)(10,17)(11,18)(13,22)(14,20)(15,23)(16,24)(19,21)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,40)(34,35);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,21)(13,17)(14,19)(16,22)(18,20)(23,24)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,35,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,17)(12,22)(14,23)(15,20)(16,21)(19,24)(25,29,30,28)(26,33,34,32)(27,37,38,36)(31,40,35,39)Polycyclic presentation:
2^9|b^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9422728);