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 | C23×D4 | 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18)(19,25)(20,21)(22,26)(23,29)(24,31)(27,28)(30,32)(33,34,39,37)(35,40,36,43)(38,45,46,41)(42,48,44,47);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(17,19)(18,23)(20,27)(21,30)(22,29)(24,32)(25,26)(28,31)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(3,12)(6,15)(8,16)(10,13)(17,18,22,26)(19,23,29,25)(20,21,24,31)(27,30,32,28)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,20)(18,21)(19,27)(22,24)(23,30)(25,28)(26,31)(29,32)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,21)(18,24)(19,28)(20,26)(22,31)(23,27)(25,32)(29,30)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,19)(18,25)(20,27)(21,28)(22,29)(23,26)(24,32)(30,31)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8314251);