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 | |
4006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(9,14)(10,15)(11,16)(17,18,22,23)(19,26,21,24)(20,28,29,25)(27,31,30,32)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(1,2)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(18,23)(20,29)(24,26)(27,30)(34,39)(35,41)(40,46)(42,47);(2,7)(3,8)(9,14)(10,15)(17,19,22,21)(18,24,23,26)(20,30,29,27)(25,32,28,31)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,3,6,8)(2,4,7,12)(5,10,13,15)(9,11,14,16)(17,20,22,29)(18,25,23,28)(19,27,21,30)(24,31,26,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32);(1,5)(2,9)(3,10)(4,11)(6,13)(7,14)(8,15)(12,16)(17,21,22,19)(18,26,23,24)(20,30,29,27)(25,32,28,31)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019754);