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 | |
4291 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)(3,8)(4,6)(5,7)(9,10,15,19)(11,16,20,24)(12,14,18,23)(13,22,21,17)(25,26,32,36)(27,37,31,33)(28,39,38,34)(29,40,35,30);(1,3,6,5)(2,7,4,8)(11,20)(13,21)(16,24)(17,22)(25,27,32,31)(26,33,36,37)(28,35,38,29)(30,39,40,34);(9,11,15,20)(10,16,19,24)(12,21,18,13)(14,17,23,22)(25,28)(26,34)(27,29)(30,37)(31,35)(32,38)(33,40)(36,39);(1,2,6,4)(3,7,5,8)(9,12)(10,14)(11,21)(13,20)(15,18)(16,17)(19,23)(22,24)(25,29,32,35)(26,30,36,40)(27,38,31,28)(33,34,37,39);(1,4,6,2)(3,7,5,8)(9,13,15,21)(10,17,19,22)(11,12,20,18)(14,16,23,24)(25,30)(26,35)(27,39)(28,37)(29,36)(31,34)(32,40)(33,38);(1,5,6,3)(2,7,4,8)(9,14)(10,18)(11,17)(12,19)(13,16)(15,23)(20,22)(21,24)(25,31,32,27)(26,33,36,37)(28,35,38,29)(30,34,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=hi,d^2=g,e^2=gi,f^2=g,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, 8399989);