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 | |
4049 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)(3,8)(4,10)(5,6)(7,12)(9,13)(11,16)(14,15)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,34,38,37)(35,39,44,42)(36,40,45,46)(41,47,43,48);(1,3,7,14)(2,8,12,15)(4,11,13,5)(6,10,16,9)(17,18,21,24)(19,22,27,25)(20,23,29,28)(26,30,32,31)(33,35)(34,39)(36,43)(37,42)(38,44)(40,48)(41,45)(46,47);(1,4,7,13)(2,9,12,10)(3,5,14,11)(6,8,16,15)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,36,38,45)(34,40,37,46)(35,41,44,43)(39,47,42,48);(1,4)(2,10)(3,11)(5,14)(6,15)(7,13)(8,16)(9,12)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,37,38,34)(35,42,44,39)(36,40,45,46)(41,47,43,48);(1,5,7,11)(2,6,12,16)(3,13,14,4)(8,9,15,10)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,36)(34,40)(35,43)(37,46)(38,45)(39,48)(41,44)(42,47);(1,6)(2,11)(3,10)(4,15)(5,12)(7,16)(8,13)(9,14)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(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=gh,f^2=g,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, 8900181);