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 | |
4485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,18)(16,20)(22,24)(25,26,30,31)(27,28,35,32)(29,37,38,33)(34,39,40,36);(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,17,18)(12,20,21,16)(19,24,23,22)(26,31)(28,32)(29,38)(34,40);(2,5)(6,7)(11,17)(15,18)(19,23)(22,24)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,39,37,36);(2,5)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,27)(29,36)(30,32)(31,35)(33,34)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10228112);