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 | |
3882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,14,13)(4,15,5,9)(6,10,16,12)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,34)(35,42)(36,46)(37,45)(38,40)(39,41)(43,47)(44,48);(1,3,7,14)(2,8,11,13)(4,12,5,10)(6,9,16,15)(19,25)(23,27)(26,31)(30,32)(33,35,40,45)(34,37,38,42)(36,43,39,44)(41,47,46,48);(1,4,7,5)(2,9,11,15)(3,10,14,12)(6,13,16,8)(17,19,22,25)(18,23,21,27)(20,26,28,31)(24,30,29,32)(33,36,40,39)(34,41,38,46)(35,43,45,44)(37,47,42,48);(1,5,7,4)(2,9,11,15)(3,12,14,10)(6,13,16,8)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37)(34,35)(36,47)(38,45)(39,48)(40,42)(41,43)(44,46);(1,2)(3,13)(4,15)(5,9)(6,10)(7,11)(8,14)(12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,38,40,34)(35,37,45,42)(36,41,39,46)(43,48,44,47);(1,6,7,16)(2,10,11,12)(3,9,14,15)(4,13,5,8)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,39,40,36)(34,41,38,46)(35,44,45,43)(37,47,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8547070);