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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(18,24)(20,25)(26,31)(28,32)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,2)(3,8)(4,5)(6,7)(9,14)(10,16)(11,13)(12,15)(17,18,23,24)(19,25,21,20)(22,26,30,31)(27,32,29,28)(35,41)(38,45)(42,47)(46,48);(2,7)(5,9)(8,15)(11,16)(17,19,23,21)(18,25,24,20)(22,29,30,27)(26,28,31,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,8)(4,13)(5,11)(6,12)(7,15)(9,16)(10,14)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,31)(30,32)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,4)(2,9)(3,10)(5,7)(6,14)(8,11)(12,13)(15,16)(17,21,23,19)(18,25,24,20)(22,29,30,27)(26,32,31,28)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,5,6,9)(2,4,7,14)(3,11,12,16)(8,13,15,10)(17,22,23,30)(18,26,24,31)(19,27,21,29)(20,28,25,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8279318);