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 | |
4541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,16)(12,20)(13,18)(14,15)(17,21)(19,24)(22,23)(25,26,30,33)(27,34,31,28)(29,32,38,40)(35,39,37,36);(2,5)(6,7)(10,15)(12,21)(16,24)(18,22)(27,31)(28,34)(35,37)(36,39);(1,2)(3,6)(4,5)(7,8)(10,15)(13,23)(16,24)(17,20)(25,27,30,31)(26,28,33,34)(29,37,38,35)(32,39,40,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,17)(14,19)(15,24)(18,21)(20,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,3,4,8)(2,6,5,7)(9,12,14,21)(10,17,15,20)(11,18,19,22)(13,16,23,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,37)(35,40)(38,39);(9,13,14,23)(10,18,15,22)(11,17,19,20)(12,16,21,24)(25,29,30,38)(26,32,33,40)(27,35,31,37)(28,36,34,39)Polycyclic presentation:
2^9|a^2=i,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325576);