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 | |
4409 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,17)(11,14,20,16)(12,23,22,18)(13,24,21,19)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,21)(13,22)(14,17)(15,20)(18,19)(23,24);(10,17)(11,20)(12,22)(19,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,3,5,4)(2,7,6,8)(9,12,15,22)(10,18,17,23)(11,21,20,13)(14,24,16,19)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,13,15,21)(10,19,17,24)(11,22,20,12)(14,23,16,18)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4,5,3)(2,8,6,7)(9,14)(10,20)(11,17)(12,24)(13,18)(15,16)(19,22)(21,23)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7824483);