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 | |
4426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(6,7)(9,10)(11,13)(12,15)(14,20)(16,18)(17,22)(19,23)(21,24)(25,26,32,36)(27,37,40,33)(28,29,30,34)(31,39,38,35);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(25,27)(26,33)(28,35)(29,38)(30,39)(31,34)(32,40)(36,37);(1,3)(2,6)(4,7)(5,8)(10,12)(11,18)(17,20)(19,24)(25,28,32,30)(26,34,36,29)(27,35,40,39)(31,37,38,33);(1,2,5,4)(3,7,8,6)(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,29,32,34)(26,28,36,30)(27,38,40,31)(33,35,37,39);(1,4,5,2)(3,7,8,6)(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,30,32,28)(26,34,36,29)(27,39,40,35)(31,37,38,33);(1,3)(2,7)(4,6)(5,8)(9,14,15,22)(10,17,12,20)(11,19,18,24)(13,21,16,23)(25,31,32,38)(26,35,36,39)(27,34,40,29)(28,37,30,33)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=hi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9758341);