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 | |
4146 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)(6,11)(7,8)(10,15)(12,14)(13,16)(33,34,39,40)(35,42,36,41)(37,46,47,43)(38,44,48,45);(1,3,7,10)(2,5,8,15)(4,12,13,6)(9,14,16,11)(17,18)(19,26)(20,21)(22,23)(24,29)(25,31)(27,28)(30,32)(34,40)(37,47)(41,42)(44,45);(2,8)(3,10)(6,12)(9,16)(18,23)(21,25)(24,26)(28,32)(33,35,39,36)(34,41,40,42)(37,45,47,44)(38,46,48,43);(1,4)(2,9)(3,12)(5,14)(6,10)(7,13)(8,16)(11,15)(17,19,22,29)(18,24,23,26)(20,30,31,27)(21,28,25,32)(33,36,39,35)(34,42,40,41)(37,45,47,44)(38,46,48,43);(1,5,7,15)(2,10,8,3)(4,11,13,14)(6,9,12,16)(17,20)(18,25)(19,27)(21,23)(22,31)(24,28)(26,32)(29,30)(33,37)(34,43)(35,45)(36,44)(38,41)(39,47)(40,46)(42,48);(1,6,7,12)(2,11,8,14)(3,4,10,13)(5,9,15,16)(17,21,22,25)(18,20,23,31)(19,28,29,32)(24,30,26,27)(33,38,39,48)(34,44,40,45)(35,43,36,46)(37,41,47,42)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8309496);