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 | |
3973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,13,7)(4,14,10,5)(11,12,16,15)(17,18,23,21)(19,26,24,20)(22,25,30,29)(27,32,31,28)(33,34,39,42)(35,37,41,47)(36,38,46,40)(43,48,44,45);(1,3,6,13)(2,7,8,9)(4,11,10,16)(5,12,14,15)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,32)(30,31)(33,35,39,41)(34,37,42,47)(36,44,46,43)(38,45,40,48);(2,8)(4,10)(7,9)(11,16)(33,36,39,46)(34,40,42,38)(35,43,41,44)(37,45,47,48);(1,4,6,10)(2,5,8,14)(3,11,13,16)(7,12,9,15)(17,20,23,26)(18,24,21,19)(22,28,30,32)(25,31,29,27)(33,36)(34,38)(35,44)(37,45)(39,46)(40,42)(41,43)(47,48);(1,3)(2,9)(4,11)(5,15)(6,13)(7,8)(10,16)(12,14)(17,21,23,18)(19,26,24,20)(22,25,30,29)(27,28,31,32)(33,37,39,47)(34,41,42,35)(36,45,46,48)(38,43,40,44);(1,5)(2,10)(3,12)(4,8)(6,14)(7,16)(9,11)(13,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38)(34,36)(35,45)(37,44)(39,40)(41,48)(42,46)(43,47)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592706);