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 | |
4516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,7,5)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,30,32)(27,31,35,39)(28,33,37,40)(29,34,36,38);(1,3)(2,5)(4,8)(6,7)(10,11)(12,19)(16,18)(20,24)(25,27)(26,31)(28,29)(30,35)(32,39)(33,34)(36,37)(38,40);(2,4)(5,8)(12,19)(13,21)(15,17)(16,18)(26,32)(27,35)(29,36)(33,40);(2,4)(3,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,30,37)(26,33,32,40)(27,36,35,29)(31,38,39,34);(1,4,6,2)(3,8,7,5)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,30,36)(26,34,32,38)(27,28,35,37)(31,33,39,40);(1,5,6,8)(2,7,4,3)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947920);