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 | |
3872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,16)(4,8,14,13)(5,9,15,12)(17,18,23,24)(19,21,25,20)(22,26,31,32)(27,29,30,28)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(3,11)(4,14)(7,16)(8,13)(18,24)(20,21)(26,32)(28,29)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(1,3)(2,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19)(18,20)(21,24)(22,30)(23,25)(26,29)(27,31)(28,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(17,20,23,21)(18,25,24,19)(22,28,31,29)(26,30,32,27)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,21,23,20)(18,19,24,25)(22,29,31,28)(26,27,32,30)(33,37,38,45)(34,40,41,47)(35,42,43,48)(36,44,39,46);(1,5)(2,9)(3,13)(4,7)(6,15)(8,11)(10,12)(14,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109443);