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 | |
4545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,13)(11,17,18,15)(12,16,20,22)(19,24,21,23)(25,26)(27,32)(28,38)(29,39)(30,33)(31,37)(34,40)(35,36);(1,2)(3,7)(4,5)(6,8)(10,13)(11,18)(16,22)(19,21)(25,27)(26,31)(28,35)(29,36)(30,37)(32,33)(34,39)(38,40);(2,5)(6,7)(12,20)(16,22)(19,21)(23,24)(25,28)(26,29)(27,34)(30,39)(31,40)(32,36)(33,38)(35,37);(2,5)(6,7)(9,11,14,18)(10,15,13,17)(12,21,20,19)(16,24,22,23)(25,28)(26,29)(27,35)(30,39)(31,36)(32,40)(33,38)(34,37);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,16,13,22)(11,19,18,21)(15,23,17,24)(25,27)(26,32)(28,34)(29,36)(30,37)(31,33)(35,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,17,18,15)(12,22,20,16)(19,24,21,23)(25,29)(26,28)(27,36)(30,38)(31,35)(32,34)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,d^2=gh,e^2=gh,f^2=gh,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, 10231286);