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 | |
4130 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,16)(12,15)(18,21)(19,25)(24,28)(26,31)(33,34)(35,43)(36,47)(37,48)(38,44)(39,41)(40,46)(42,45);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,18,22,21)(19,23,25,27)(20,24,29,28)(26,30,31,32)(33,35,39,46)(34,40,41,43)(36,42,48,44)(37,45,47,38);(2,7)(3,8)(9,13)(11,15)(19,25)(23,27)(26,31)(30,32)(34,41)(36,48)(40,43)(42,44);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,19,22,25)(18,23,21,27)(20,26,29,31)(24,30,28,32)(33,36)(34,37)(35,44)(38,40)(39,48)(41,47)(42,46)(43,45);(1,4,6,10)(2,8,7,3)(5,12,14,16)(9,15,13,11)(17,20,22,29)(18,24,21,28)(19,26,25,31)(23,30,27,32)(33,37)(34,36)(35,45)(38,46)(39,47)(40,42)(41,48)(43,44);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,22,18)(19,27,25,23)(20,28,29,24)(26,32,31,30)(33,38,39,45)(34,42,41,44)(35,37,46,47)(36,40,48,43)Polycyclic presentation:
2^9|b^2=g,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014408);