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 | |
3971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,10,4,6)(5,13,15,8)(11,14,12,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(35,43)(36,45)(39,46)(40,48);(1,2,7,9)(3,10,4,6)(5,8,15,13)(11,16,12,14)(18,23)(19,26)(25,28)(27,32)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,44,47);(1,3,7,4)(2,6,9,10)(5,11,15,12)(8,14,13,16)(17,18,22,23)(19,24,26,21)(20,25,30,28)(27,31,32,29)(33,35)(34,39)(36,44)(37,45)(38,43)(40,47)(41,48)(42,46);(1,4,7,3)(2,6,9,10)(5,12,15,11)(8,14,13,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,7,15)(2,8,9,13)(3,11,4,12)(6,14,10,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36,38,45)(34,40,42,48)(35,44,43,37)(39,47,46,41);(1,6)(2,3)(4,9)(5,14)(7,10)(8,11)(12,13)(15,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,37,38,44)(34,41,42,47)(35,45,43,36)(39,48,46,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=i,d^2=gi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8104299);