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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(18,21)(19,20)(26,29)(27,28)(33,34,39,40)(35,37,42,38)(36,44,47,41)(43,46,48,45);(1,2,6,7)(3,5,9,14)(4,8,13,15)(10,12,11,16)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(34,40)(35,42)(41,44)(43,48);(2,7)(5,14)(8,15)(12,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35)(34,37)(36,43)(38,40)(39,42)(41,45)(44,46)(47,48);(1,3,6,9)(2,5,7,14)(4,11,13,10)(8,16,15,12)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,4,6,13)(2,8,7,15)(3,10,9,11)(5,12,14,16)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,37,39,38)(34,42,40,35)(36,45,47,46)(41,48,44,43);(1,5)(2,9)(3,7)(4,12)(6,14)(8,11)(10,15)(13,16)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,38,39,37)(34,35,40,42)(36,46,47,45)(41,43,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8224724);