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 | |
4428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,20)(12,14)(13,16)(15,17)(18,19)(21,24)(22,23)(25,26,31,30)(27,34,32,28)(29,33,38,39)(35,40,37,36);(2,3)(6,7)(9,11,15,13)(10,16,17,20)(12,21,19,23)(14,22,18,24)(25,27)(26,28)(29,37)(30,34)(31,32)(33,40)(35,38)(36,39);(1,2,5,3)(4,6,8,7)(10,17)(12,19)(16,20)(21,23);(9,12,15,19)(10,18,17,14)(11,21,13,23)(16,24,20,22)(25,28,31,34)(26,32,30,27)(29,36,38,40)(33,37,39,35);(1,3,5,2)(4,6,8,7)(9,13,15,11)(10,16,17,20)(12,23,19,21)(14,22,18,24)(25,29)(26,33)(27,35)(28,36)(30,39)(31,38)(32,37)(34,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,18)(10,19,17,12)(11,22,13,24)(16,23,20,21)(25,30,31,26)(27,34,32,28)(29,33,38,39)(35,36,37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=i,d^2=g,e^2=hi,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, 9755736);