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 |
| prime radical | #49634 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,5,11,7)(4,8,12,15)(10,13,16,14)(17,18,22,26)(19,23,28,32)(20,29,30,24)(21,25,27,31)(33,34,37,40)(35,38,43,41)(36,39,45,48)(42,46,44,47);(4,12)(8,15)(10,16)(13,14)(17,19,22,28)(18,23,26,32)(20,27,30,21)(24,25,29,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,6,11)(2,7,9,5)(4,10,12,16)(8,14,15,13)(17,20,22,30)(18,24,26,29)(19,21,28,27)(23,31,32,25);(1,3)(2,5)(4,10)(6,11)(7,9)(8,13)(12,16)(14,15)(17,20)(18,24)(19,27)(21,28)(22,30)(23,25)(26,29)(31,32);(1,4,6,12)(2,8,9,15)(3,10,11,16)(5,13,7,14)(17,21,22,27)(18,25,26,31)(19,20,28,30)(23,29,32,24)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(1,5)(2,3)(4,13)(6,7)(8,10)(9,11)(12,14)(15,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,27)(28,32)(29,30)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,e^2=h,f^2=g,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, 9820859);