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 | |
4392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,15)(11,16,18,12)(13,17,22,23)(19,21,24,20)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,37)(38,40);(10,15)(11,18)(17,23)(19,24)(25,27,31,30)(26,32,28,34)(29,35,37,38)(33,39,36,40);(1,2,5,4)(3,6,7,8)(25,26,31,28)(27,34,30,32)(29,33,37,36)(35,40,38,39);(3,7)(6,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,21)(20,23)(22,24)(25,28,31,26)(27,32,30,34)(29,33,37,36)(35,40,38,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,11,15,18)(13,21,22,20)(17,24,23,19)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40);(1,4,5,2)(3,8,7,6)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)(25,30,31,27)(26,32,28,34)(29,35,37,38)(33,40,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=g,f^2=h,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, 10395534);