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 | |
3943 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)(3,9)(4,7)(5,10)(6,11)(8,14)(12,15)(13,16)(17,18,23,21)(19,24,27,20)(22,25,31,29)(26,30,32,28)(33,34,40,44)(35,39,37,42)(36,43,38,41)(45,48,46,47);(1,3)(2,5)(4,12)(6,10)(7,13)(8,15)(9,11)(14,16)(17,19,23,27)(18,24,21,20)(22,26,31,32)(25,30,29,28)(33,35)(34,39)(36,46)(37,40)(38,45)(41,48)(42,44)(43,47);(1,4,6,14)(2,7,11,8)(3,12,10,16)(5,13,9,15)(17,20,23,24)(18,19,21,27)(22,28,31,30)(25,26,29,32)(33,36,40,38)(34,41,44,43)(35,45,37,46)(39,47,42,48);(1,4)(2,8)(3,12)(5,15)(6,14)(7,11)(9,13)(10,16)(17,20)(18,19)(21,27)(22,30)(23,24)(25,32)(26,29)(28,31)(33,37,40,35)(34,42,44,39)(36,45,38,46)(41,47,43,48);(1,3)(2,9)(4,12)(5,11)(6,10)(7,15)(8,13)(14,16)(17,21,23,18)(19,20,27,24)(22,25,31,29)(26,30,32,28)(33,38,40,36)(34,43,44,41)(35,45,37,46)(39,47,42,48);(1,5,6,9)(2,10,11,3)(4,13,14,15)(7,16,8,12)(17,22,23,31)(18,25,21,29)(19,26,27,32)(20,28,24,30)(33,39,40,42)(34,37,44,35)(36,47,38,48)(41,46,43,45)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gi,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8633810);