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 | |
4438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,15,13)(11,18,14,16)(12,21,22,17)(19,23,20,24)(25,26,31,35)(27,36,30,32)(28,33,34,29)(37,40,39,38);(1,3,4,8)(2,5,6,7)(9,11,15,14)(10,16,13,18)(12,19,22,20)(17,23,21,24)(25,27,31,30)(26,32,35,36)(28,37,34,39)(29,40,33,38);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,28,31,34)(26,33,35,29)(27,37,30,39)(32,38,36,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,20)(13,21)(14,19)(15,22)(16,24)(18,23)(25,26)(27,32)(28,33)(29,34)(30,36)(31,35)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,13,15,10)(11,16,14,18)(12,21,22,17)(19,23,20,24)(25,29,31,33)(26,34,35,28)(27,38,30,40)(32,37,36,39);(9,14,15,11)(10,16,13,18)(12,20,22,19)(17,23,21,24)(25,30,31,27)(26,32,35,36)(28,39,34,37)(29,40,33,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8412926);