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 | |
4508 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)(25,26,32,35)(27,36,30,33)(28,31,29,34)(37,38,39,40);(10,15)(11,18)(17,23)(19,24)(25,27,32,30)(26,33,35,36)(28,39,29,37)(31,38,34,40);(1,2,5,4)(3,6,7,8)(25,28,32,29)(26,34,35,31)(27,37,30,39)(33,38,36,40);(3,7)(6,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,21)(20,23)(22,24)(25,29,32,28)(26,31,35,34)(27,37,30,39)(33,38,36,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,11,15,18)(13,21,22,20)(17,24,23,19)(25,30,32,27)(26,33,35,36)(28,39,29,37)(31,40,34,38);(1,4,5,2)(3,8,7,6)(9,13,14,22)(10,17,15,23)(11,19,18,24)(12,20,16,21)(25,31)(26,28)(27,38)(29,35)(30,40)(32,34)(33,39)(36,37)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=g,f^2=gh,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, 10395557);