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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4220 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)(3,5)(4,9)(6,16)(7,10)(8,14)(11,13)(12,15)(17,18,22,24)(19,25,27,21)(20,23,30,28)(26,31,32,29)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,3,7,6)(2,5,10,16)(4,12,14,11)(8,13,9,15)(17,19)(18,21)(20,26)(22,27)(23,29)(24,25)(28,31)(30,32)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(1,4,7,14)(2,8,10,9)(3,11,6,12)(5,15,16,13)(17,18,22,24)(19,21,27,25)(20,28,30,23)(26,31,32,29)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,10)(13,16)(17,18)(19,21)(20,28)(22,24)(23,30)(25,27)(26,31)(29,32)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,5)(2,3)(4,13)(6,10)(7,16)(8,12)(9,11)(14,15)(17,20,22,30)(18,23,24,28)(19,26,27,32)(21,29,25,31)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,6,7,3)(2,5,10,16)(4,11,14,12)(8,13,9,15)(17,21)(18,19)(20,29)(22,25)(23,26)(24,27)(28,32)(30,31)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gi,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900917);