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 | |
4164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,15,14)(5,13,16,10)(17,18,23,25)(19,24,21,27)(20,26,31,30)(22,29,32,28)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(2,7)(4,15)(8,11)(10,13)(17,19)(18,24)(20,29)(21,23)(22,30)(25,27)(26,32)(28,31)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,7)(3,12)(4,15)(5,16)(18,25)(19,21)(20,31)(22,32)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,3)(2,8)(4,10)(5,14)(6,12)(7,11)(9,16)(13,15)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,24,32,27)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,4)(2,9)(3,13)(5,8)(6,15)(7,14)(10,12)(11,16)(17,21,23,19)(18,27,25,24)(20,29,31,28)(22,26,32,30)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,5)(2,10)(3,14)(4,8)(6,16)(7,13)(9,12)(11,15)(17,22,23,32)(18,28,25,29)(19,30,21,26)(20,24,31,27)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=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, 9405652);