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 | |
4325 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,16)(13,17)(14,18)(15,19)(21,22)(23,24)(26,30)(28,35)(31,36)(34,39);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,16,19,20)(13,21,18,23)(14,22,17,24)(27,33)(31,36)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,16,19,20)(13,23,18,21)(14,22,17,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,13)(10,17)(11,21)(12,23)(14,19)(15,18)(16,24)(20,22)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40);(9,14,15,17)(10,18,19,13)(11,22,12,24)(16,23,20,21)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8595532);