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 | |
4260 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)(5,15)(6,7)(8,12)(10,16)(13,14)(18,23)(19,26)(25,30)(27,32)(33,34,39,45)(35,40,47,48)(36,42,46,41)(37,43,44,38);(2,7)(4,14)(8,11)(10,15)(17,18,22,23)(19,24,26,20)(21,25,29,30)(27,31,32,28)(33,35,39,47)(34,40,45,48)(36,44,46,37)(38,41,43,42);(2,7)(3,12)(4,14)(5,16)(19,26)(20,24)(21,29)(25,30)(33,36,39,46)(34,41,45,42)(35,37,47,44)(38,48,43,40);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,45)(43,48);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32)(33,37,39,44)(34,43,45,38)(35,46,47,36)(40,41,48,42);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,38,39,43)(34,44,45,37)(35,41,47,42)(36,40,46,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8922522);