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 | |
4567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,12,15,16)(18,23)(19,27)(21,33)(25,35)(29,41)(31,34)(38,44)(47,48);(2,5)(6,7)(10,14)(11,15)(18,23)(24,28)(25,35)(26,36)(37,43)(38,44)(39,40)(47,48);(9,11)(10,12)(13,15)(14,16)(17,18)(19,28)(20,25)(21,26)(22,23)(24,27)(29,43)(30,44)(31,39)(32,35)(33,36)(34,40)(37,41)(38,42)(45,48)(46,47);(9,12)(10,11)(13,16)(14,15)(17,19)(18,24)(20,29)(21,30)(22,27)(23,28)(25,37)(26,38)(31,45)(32,41)(33,42)(34,46)(35,43)(36,44)(39,47)(40,48);(1,2)(3,7)(4,5)(6,8)(17,20,22,32)(18,25,23,35)(19,29,27,41)(21,34,33,31)(24,37,28,43)(26,40,36,39)(30,46,42,45)(38,48,44,47);(1,3)(2,6)(4,8)(5,7)(17,21,22,33)(18,26,23,36)(19,30,27,42)(20,31,32,34)(24,38,28,44)(25,39,35,40)(29,45,41,46)(37,47,43,48)Polycyclic presentation:
2^9|a^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820099);