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 | |
4602 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)(3,4)(5,6)(7,8)(9,10,14,17)(11,15,19,13)(12,16,21,24)(18,20,23,22)(25,26)(27,31)(28,32)(29,35)(30,33)(34,37)(36,39)(38,40);(2,6)(3,7)(9,11,14,19)(10,13,17,15)(12,18,21,23)(16,22,24,20)(25,27,30,35)(26,31,33,29)(28,34,38,36)(32,37,40,39);(1,3,5,7)(2,4,6,8)(25,27)(26,29)(28,34)(30,35)(31,33)(32,39)(36,38)(37,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,20)(21,23)(22,24)(25,27)(26,29)(28,36)(30,35)(31,33)(32,37)(34,38)(39,40);(9,12)(10,16)(11,18)(13,22)(14,21)(15,20)(17,24)(19,23)(25,28)(26,32)(27,34)(29,39)(30,38)(31,37)(33,40)(35,36);(9,13,14,15)(10,11,17,19)(12,20,21,22)(16,23,24,18)(25,29)(26,27)(28,37)(30,31)(32,36)(33,35)(34,40)(38,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10441561);