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 |
| prime radical | #2819 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,19)(11,16,22,15)(12,13,18,17)(20,23,24,21)(25,26,30,31)(27,32,34,29)(28,36,38,33)(35,37,40,39);(2,3)(6,7)(9,10)(11,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(26,31)(27,34)(33,36)(35,40);(1,2,5,3)(4,6,8,7)(9,11,14,22)(10,15,19,16)(12,20,18,24)(13,21,17,23)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,20)(13,23)(14,22)(15,19)(17,21)(18,24)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,4)(2,6)(3,7)(5,8)(9,12)(10,17)(11,20)(13,19)(14,18)(15,23)(16,21)(22,24)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40);(9,13,14,17)(10,18,19,12)(11,21,22,23)(15,24,16,20)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658006);