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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4114 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,22,26,29)(20,24,28,31)(25,30,27,32)(33,34)(35,38)(36,43)(37,40)(39,45)(41,48)(42,47)(44,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,21,26)(18,22,23,29)(20,27,28,25)(24,32,31,30)(33,35,37,42)(34,38,40,47)(36,44,45,41)(39,48,43,46);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(18,23)(22,29)(24,31)(30,32)(33,36)(34,39)(35,41)(37,45)(38,46)(40,43)(42,44)(47,48);(2,7)(6,9)(8,14)(13,15)(17,20,21,28)(18,24,23,31)(19,25,26,27)(22,30,29,32)(35,42)(36,45)(38,47)(39,43);(3,10)(6,9)(11,16)(13,15)(18,23)(22,29)(24,31)(30,32)(34,40)(35,42)(39,43)(41,44);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=hi,f^2=g,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, 8011460);