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×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,15)(11,14)(12,16)(17,18,23,27)(19,26,22,24)(20,21,25,32)(28,31,30,29)(33,34,38,42)(35,41,37,39)(36,44,46,40)(43,47,45,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,19,23,22)(18,24,27,26)(20,28,25,30)(21,29,32,31)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,48)(45,47);(1,3,6,7)(2,4,9,10)(5,11,13,14)(8,12,15,16)(17,20,23,25)(18,21,27,32)(19,28,22,30)(24,29,26,31)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,24)(20,21)(22,26)(23,27)(25,32)(28,29)(30,31)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,4)(2,7)(3,9)(5,12)(6,10)(8,14)(11,15)(13,16)(17,21)(18,25)(19,29)(20,27)(22,31)(23,32)(24,30)(26,28)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,48)(45,47);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,22,23,19)(18,26,27,24)(20,30,25,28)(21,31,32,29)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044776);