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 | |
4176 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(2,7)(3,11)(4,14)(13,15)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(33,35,38,43)(34,36,40,41)(37,42,46,48)(39,44,45,47);(2,7)(4,14)(8,10)(12,16)(18,23)(20,29)(24,30)(26,32)(33,36,38,41)(34,35,40,43)(37,44,46,47)(39,42,45,48);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,13)(14,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,34)(35,36)(37,45)(38,40)(39,46)(41,43)(42,47)(44,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8525331);