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×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(8,14)(10,15)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,29)(30,32)(36,44)(39,46)(42,48)(45,47);(4,11)(8,14)(10,15)(12,16)(17,19,22,24)(18,21,25,26)(20,28,30,27)(23,31,32,29)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,31)(30,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,20,22,30)(18,23,25,32)(19,27,24,28)(21,29,26,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21)(18,24)(19,25)(20,29)(22,26)(23,28)(27,32)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=hi,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441703);