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 | |
4172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,12)(3,13,5,8)(4,10,15,9)(6,14,16,11)(18,23)(19,25)(21,29)(28,31)(34,39)(35,40)(37,45)(43,46);(1,3,7,5)(2,8,12,13)(4,11,15,14)(6,10,16,9)(17,18,22,23)(19,20,25,24)(21,26,29,30)(27,28,32,31)(34,39)(35,40)(41,47)(42,48);(1,4)(2,9)(3,11)(5,14)(6,8)(7,15)(10,12)(13,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,34,38,39)(35,36,40,44)(37,41,45,47)(42,43,48,46);(1,4)(2,10)(3,14)(5,11)(6,8)(7,15)(9,12)(13,16)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,32)(30,31)(33,35)(34,36)(37,42)(38,40)(39,44)(41,43)(45,48)(46,47);(1,5,7,3)(2,8,12,13)(4,14,15,11)(6,10,16,9)(17,20)(18,25)(19,23)(21,28)(22,24)(26,32)(27,30)(29,31)(33,36)(34,40)(35,39)(37,46)(38,44)(41,42)(43,45)(47,48);(1,6,7,16)(2,11,12,14)(3,10,5,9)(4,8,15,13)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,37,38,45)(34,41,39,47)(35,42,40,48)(36,43,44,46)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10413522);