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 | |
4150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28)(33,34,38,39)(35,40,43,36)(37,45,47,41)(42,44,48,46);(4,11)(8,14)(10,15)(12,16)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(34,39)(36,40)(41,45)(44,46);(2,7)(3,9)(8,14)(10,15)(18,23)(19,25)(24,32)(26,27)(33,35,38,43)(34,36,39,40)(37,42,47,48)(41,44,45,46);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,21)(20,24)(22,23)(25,31)(26,30)(27,28)(29,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,46)(44,45)(47,48);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,20,22,29)(18,24,23,32)(19,26,25,27)(21,30,31,28)(33,36,38,40)(34,35,39,43)(37,46,47,44)(41,48,45,42);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)(33,37,38,47)(34,41,39,45)(35,42,43,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=h,f^2=g,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, 9619587);