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 | |
3835 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34,39,37)(35,40,43,36)(38,45,47,41)(42,44,46,48);(3,9)(7,13)(10,15)(14,16)(17,19,24,22)(18,25,23,28)(20,29,21,30)(26,31,27,32)(33,35,39,43)(34,40,37,36)(38,42,47,46)(41,48,45,44);(2,6)(7,13)(8,11)(14,16)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,36,39,40)(34,35,37,43)(38,44,47,48)(41,42,45,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,21,24,20)(18,27,23,26)(19,29,22,30)(25,31,28,32)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,48,46,44);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,22,24,19)(18,25,23,28)(20,30,21,29)(26,31,27,32)(33,35)(34,40)(36,37)(38,46)(39,43)(41,44)(42,47)(45,48);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,48)(43,46)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227561);