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 | |
3956 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,15)(13,16)(17,18,23,25)(19,24,28,32)(20,22,27,30)(21,29,31,26)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(1,3,5,11)(2,6,7,9)(4,13,14,10)(8,16,12,15)(17,19,23,28)(18,24,25,32)(20,26,27,29)(21,30,31,22)(35,43)(36,46)(39,48)(40,45);(2,7)(6,9)(8,12)(15,16)(18,25)(19,28)(22,30)(26,29)(33,35)(34,39)(36,44)(37,40)(38,43)(41,46)(42,48)(45,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,15)(12,16)(13,14)(17,20,23,27)(18,22,25,30)(19,29,28,26)(21,24,31,32);(1,4,5,14)(2,8,7,12)(3,10,11,13)(6,15,9,16)(17,21)(18,26)(19,22)(20,24)(23,31)(25,29)(27,32)(28,30)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,22)(18,27)(19,21)(20,25)(23,30)(24,29)(26,32)(28,31)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=gh,e^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=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109385);