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 | |
4770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,12,16,20)(13,21,22,17)(18,24,23,19)(25,26,30,33)(27,34,31,28)(29,32,38,40)(35,37,39,36);(1,3,4,7)(2,5,6,8)(9,11,14,16)(10,12,15,20)(13,18,22,23)(17,19,21,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,16)(13,22)(19,24)(25,28,30,34)(26,31,33,27)(29,36,38,37)(32,39,40,35);(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,28)(26,27)(29,37)(30,34)(31,33)(32,39)(35,40)(36,38);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,21)(22,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,40)(34,37);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,18)(12,19)(14,22)(15,21)(16,23)(20,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8834142);