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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,11,16)(4,6,14,15)(5,13,12,10)(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,44,48,41);(1,3)(2,8)(4,5)(6,13)(7,11)(9,16)(10,15)(12,14)(18,24)(19,20)(26,31)(27,28)(33,35)(34,39)(36,41)(37,40)(38,45)(42,43)(44,47)(46,48);(2,9)(3,11)(4,14)(10,13)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,36,38,47)(34,40,42,46)(35,41,45,44)(37,43,48,39);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,15)(13,16)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32)(33,36)(34,40)(35,44)(37,43)(38,47)(39,48)(41,45)(42,46);(1,5,7,12)(2,10,9,13)(3,4,11,14)(6,8,15,16)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47);(1,6)(2,4)(3,13)(5,8)(7,15)(9,14)(10,11)(12,16)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)(33,37,38,48)(34,41,42,44)(35,40,45,46)(36,43,47,39)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014438);