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 | |
4251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,5,8,7)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26)(27,32)(28,34)(29,35)(30,40)(31,33)(36,37)(38,39);(1,3)(2,7)(4,5)(6,8)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,27,31,30)(26,32,33,40)(28,36,35,38)(29,39,34,37);(1,2,6,4)(3,7,8,5)(10,15)(11,18)(16,23)(19,20)(26,33)(28,35)(32,40)(36,38);(2,4)(3,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,28,31,35)(26,34,33,29)(27,36,30,38)(32,37,40,39);(1,4,6,2)(3,7,8,5)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,29,31,34)(26,35,33,28)(27,37,30,39)(32,36,40,38);(1,5,6,7)(2,3,4,8)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,30,31,27)(26,32,33,40)(28,38,35,36)(29,39,34,37)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=gi,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227623);