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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,9,12)(5,13,15,8)(10,16,14,11)(19,25)(22,26)(27,31)(30,32)(34,37)(35,41)(40,44)(43,47);(1,3)(2,4)(5,14)(6,9)(7,12)(8,16)(10,15)(11,13)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(33,34,38,37)(35,42,41,39)(36,44,45,40)(43,46,47,48);(2,7)(3,9)(8,13)(10,14)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,35,38,41)(34,39,37,42)(36,43,45,47)(40,46,44,48);(1,3)(2,4)(5,10)(6,9)(7,12)(8,11)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,34,38,37)(35,39,41,42)(36,40,45,44)(43,46,47,48);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,14)(15,16)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,36,38,45)(34,40,37,44)(35,43,41,47)(39,46,42,48);(1,5,6,15)(2,8,7,13)(3,10,9,14)(4,11,12,16)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,37,38,34)(35,42,41,39)(36,44,45,40)(43,48,47,46)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=ghi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664361);