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 | |
3848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(10,14)(11,16)(17,18,23,20)(19,24,26,22)(21,25,31,28)(27,32,29,30)(33,34,39,38)(35,42,40,36)(37,45,46,41)(43,44,47,48);(1,2,6,7)(3,8,9,4)(5,10,13,14)(11,15,16,12)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(33,35,39,40)(34,36,38,42)(37,43,46,47)(41,44,45,48);(5,13)(10,14)(11,16)(12,15)(19,26)(22,24)(27,29)(30,32)(33,36,39,42)(34,40,38,35)(37,44,46,48)(41,47,45,43);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,20,23,18)(19,22,26,24)(21,25,31,28)(27,32,29,30)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,30,24,32)(33,37,39,46)(34,41,38,45)(35,43,40,47)(36,44,42,48);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,29)(31,32)(33,38,39,34)(35,42,40,36)(37,45,46,41)(43,48,47,44)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=h,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8278450);