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 | |
3846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,9,5,7)(4,8,13,16)(11,15,12,14)(18,20)(19,21)(22,30)(31,32)(33,34)(35,43)(36,37)(38,40)(39,42)(41,48)(44,45)(46,47);(1,3,6,5)(2,7,10,9)(4,12,13,11)(8,15,16,14)(17,18)(19,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,48,47);(1,2,6,10)(3,7,5,9)(4,8,13,16)(11,14,12,15)(17,19,23,21)(18,24,20,27)(22,26,30,29)(25,31,28,32)(33,36,39,41)(34,37,42,48)(35,44,38,46)(40,47,43,45);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,20,23,18)(19,24,21,27)(22,28,30,25)(26,31,29,32)(33,37)(34,41)(35,45)(36,42)(38,47)(39,48)(40,44)(43,46);(1,4,6,13)(2,8,10,16)(3,11,5,12)(7,14,9,15)(17,21,23,19)(18,24,20,27)(22,29,30,26)(25,31,28,32)(33,38,39,35)(34,40,42,43)(36,46,41,44)(37,47,48,45);(1,5,6,3)(2,9,10,7)(4,12,13,11)(8,15,16,14)(17,22,23,30)(18,25,20,28)(19,26,21,29)(24,31,27,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022609);