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×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,11)(4,14)(5,12)(6,8)(7,13)(9,15)(10,16)(17,18,23,25)(19,24,29,31)(20,26,27,21)(22,28,32,30);(1,3)(2,7)(4,10)(5,14)(6,13)(8,11)(9,16)(12,15)(17,19)(18,24)(20,30)(21,32)(22,26)(23,29)(25,31)(27,28)(33,34,38,39)(35,40,44,46)(36,43,45,41)(37,42,47,48);(2,8)(5,16)(7,11)(9,14)(18,25)(19,29)(21,26)(28,30)(34,39)(36,45)(40,46)(42,48);(1,4,6,15)(2,9,8,14)(3,12,13,10)(5,11,16,7)(17,20,23,27)(18,26,25,21)(19,28,29,30)(22,24,32,31)(33,35)(34,40)(36,42)(37,41)(38,44)(39,46)(43,47)(45,48);(1,5)(2,10)(3,9)(4,11)(6,16)(7,15)(8,12)(13,14)(17,21,23,26)(18,27,25,20)(19,22,29,32)(24,30,31,28)(33,36,38,45)(34,41,39,43)(35,42,44,48)(37,46,47,40);(1,3)(2,11)(4,10)(5,9)(6,13)(7,8)(12,15)(14,16)(17,22)(18,28)(19,26)(20,31)(21,29)(23,32)(24,27)(25,30)(33,37)(34,42)(35,43)(36,46)(38,47)(39,48)(40,45)(41,44)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=h,e^2=i,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, 8525489);