Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,10,13,8)(4,5,15,6)(11,14,16,12)(17,18,22,25)(19,21,27,23)(20,28,30,24)(26,29,32,31)(33,34)(35,38)(36,43)(37,40)(39,45)(41,48)(42,47)(44,46);(1,3,7,13)(2,8,9,10)(4,11,15,16)(5,12,6,14)(17,19,22,27)(18,23,25,21)(20,26,30,32)(24,29,28,31)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,4)(2,5)(3,11)(6,9)(7,15)(8,12)(10,14)(13,16)(17,20,22,30)(18,24,25,28)(19,26,27,32)(21,31,23,29)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(1,5,7,6)(2,4,9,15)(3,12,13,14)(8,11,10,16)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,32,28,26)(35,42)(36,45)(38,47)(39,43);(3,13)(8,10)(11,16)(12,14)(18,25)(19,27)(24,28)(26,32)(34,40)(35,42)(39,43)(41,44);(1,6,7,5)(2,4,9,15)(3,14,13,12)(8,11,10,16)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012744);