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 | |
3905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(17,18)(19,25)(20,23)(21,24)(22,28)(26,30)(27,31)(29,32)(33,34,39,38)(35,40,46,45)(36,42,37,41)(43,48,44,47);(3,10)(6,7)(11,16)(14,15)(18,20)(24,25)(26,28)(31,32)(33,35,39,46)(34,40,38,45)(36,43,37,44)(41,47,42,48);(1,3,5,10)(2,6,9,7)(4,11,13,16)(8,14,12,15)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,36,39,37)(34,41,38,42)(35,43,46,44)(40,47,45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,37,39,36)(34,42,38,41)(35,44,46,43)(40,48,45,47);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38,39,34)(35,45,46,40)(36,42,37,41)(43,48,44,47)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8072386);