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 | |
4161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(18,24)(20,25)(26,32)(28,30)(33,34,39,43)(35,44,46,40)(36,47,38,41)(37,48,45,42);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,18,23,24)(19,25,21,20)(22,26,31,32)(27,30,29,28)(33,35,39,46)(34,40,43,44)(36,45,38,37)(41,42,47,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,30,32,28)(33,36,39,38)(34,41,43,47)(35,45,46,37)(40,42,44,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,20)(18,19)(21,24)(22,30)(23,25)(26,29)(27,32)(28,31)(33,34)(35,44)(36,41)(37,42)(38,47)(39,43)(40,46)(45,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,21,23,19)(18,20,24,25)(22,29,31,27)(26,28,32,30)(33,37)(34,42)(35,36)(38,46)(39,45)(40,47)(41,44)(43,48);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,22,23,31)(18,26,24,32)(19,27,21,29)(20,28,25,30)(33,38,39,36)(34,41,43,47)(35,37,46,45)(40,42,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332722);