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 | |
4387 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26)(27,35)(28,30)(29,33)(31,32)(34,38)(36,37)(39,40);(2,6)(3,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(26,32)(28,38)(33,35)(36,39);(1,3)(2,4)(5,7)(6,8)(12,20)(15,22)(18,24)(21,23)(25,27,31,29)(26,33,32,35)(28,36,38,39)(30,40,34,37);(1,4)(2,3)(5,8)(6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,28)(26,34)(27,36)(29,39)(30,32)(31,38)(33,37)(35,40);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,29,31,27)(26,35,32,33)(28,36,38,39)(30,40,34,37);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,30,31,34)(26,28,32,38)(27,37,29,40)(33,39,35,36)Polycyclic presentation:
2^9|a^2=gh,c^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8229516);