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 | #2617 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3925 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)(4,9)(5,10)(6,8)(7,14)(12,16)(13,15)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(1,3,6,14)(2,7,8,11)(4,12,15,10)(5,13,16,9)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(2,8)(5,16)(7,11)(9,13)(19,28)(22,31)(24,32)(26,29)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,11,16,7)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,3)(2,7)(4,10)(5,13)(6,14)(8,11)(9,16)(12,15)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,11,15,7)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)(33,37,38,45)(34,40,41,47)(35,42,43,48)(36,44,39,46)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=h,f^2=g,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, 8314063);