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 | |
4556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26)(27,36)(28,34)(29,35)(30,32)(31,33)(37,40)(38,39);(1,3,5,6)(2,4,7,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27,31,30)(26,32,33,36)(28,37,38,35)(29,39,40,34);(10,11)(15,17)(16,18)(22,23)(26,33)(28,38)(32,36)(35,37);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,31,38)(26,34,33,39)(27,37,30,35)(29,36,40,32);(1,2)(3,4)(5,7)(6,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29)(26,35)(27,34)(28,36)(30,39)(31,40)(32,38)(33,37);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,27)(26,36,33,32)(28,35,38,37)(29,34,40,39)Polycyclic presentation:
2^9|b^2=g,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9619711);