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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3837 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)(3,5)(4,11)(6,9)(7,8)(10,13)(12,16)(14,15)(17,18,22,25)(19,23,28,21)(20,24,30,32)(26,31,27,29)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(2,8)(5,11)(9,15)(13,16)(17,19,22,28)(18,23,25,21)(20,27,30,26)(24,29,32,31)(35,37)(40,41)(43,45)(47,48);(1,3,7,4)(2,5,8,11)(6,10,14,12)(9,13,15,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,4,7,3)(2,5,8,11)(6,10,14,12)(9,16,15,13)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,36,39,34)(35,41,37,40)(38,42,46,44)(43,47,45,48);(1,5)(2,4)(3,8)(6,13)(7,11)(9,12)(10,15)(14,16)(17,19)(18,21)(20,26)(22,28)(23,25)(24,29)(27,30)(31,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,21)(18,19)(20,29)(22,23)(24,26)(25,28)(27,32)(30,31)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10329985);