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 | |
4725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(10,15)(11,17)(13,21)(20,23)(26,31)(27,32)(29,37)(35,38);(2,6)(3,7)(9,10,14,15)(11,12,17,16)(13,18,21,22)(19,20,24,23)(26,31)(27,32)(33,39)(34,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,26,30,31)(27,28,32,36)(29,33,37,39)(34,35,40,38);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,24)(22,23)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,40)(38,39);(9,12)(10,17)(11,15)(13,20)(14,16)(18,24)(19,22)(21,23)(25,28)(26,32)(27,31)(29,38)(30,36)(33,34)(35,37)(39,40);(9,13,14,21)(10,18,15,22)(11,19,17,24)(12,20,16,23)(25,29)(26,33)(27,34)(28,35)(30,37)(31,39)(32,40)(36,38)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=gi,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, 10441539);