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 | |
4056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,10,5)(4,8,12,16)(11,15,13,14)(17,18)(19,26)(20,28)(21,25)(22,23)(24,29)(27,31)(30,32)(33,34,38,42)(35,36,40,39)(37,45,47,41)(43,48,46,44);(1,3)(2,7)(4,11)(5,9)(6,10)(8,15)(12,13)(14,16)(17,19,23,25)(18,21,22,26)(20,27,29,32)(24,31,28,30)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(3,10)(4,12)(5,7)(8,16)(18,22)(20,29)(21,26)(27,32)(33,35,38,40)(34,39,42,36)(37,46,47,43)(41,44,45,48);(1,3)(2,7)(4,13)(5,9)(6,10)(8,14)(11,12)(15,16)(17,20,23,29)(18,24,22,28)(19,27,25,32)(21,31,26,30)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,21,23,26)(18,25,22,19)(20,30,29,31)(24,27,28,32)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,5,6,7)(2,3,9,10)(4,14,12,15)(8,11,16,13)(17,22,23,18)(19,26,25,21)(20,28,29,24)(27,30,32,31)(33,37,38,47)(34,41,42,45)(35,43,40,46)(36,44,39,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=gi,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, 8927198);