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 | |
4107 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)(3,9,11,5)(4,12,14,7)(10,13,16,15)(18,23)(20,24)(25,31)(28,32)(34,39)(35,42)(36,44)(45,47);(1,3,6,11)(2,5,8,9)(4,10,14,16)(7,13,12,15)(18,23)(19,26)(25,31)(27,29)(33,34,38,39)(35,37,42,40)(36,41,44,46)(43,45,48,47);(1,2)(3,5)(4,7)(6,8)(9,11)(10,13)(12,14)(15,16)(17,18,22,23)(19,24,26,20)(21,25,30,31)(27,32,29,28)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,2)(3,9)(4,7)(5,11)(6,8)(10,15)(12,14)(13,16)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,30)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4,6,14)(2,7,8,12)(3,10,11,16)(5,13,9,15)(17,20,22,24)(18,19,23,26)(21,28,30,32)(25,27,31,29)(33,36,38,44)(34,41,39,46)(35,43,42,48)(37,45,40,47);(1,5,6,9)(2,3,8,11)(4,13,14,15)(7,10,12,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=i,e^2=h,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, 8332706);