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 | |
3854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34,39,42)(35,41,36,40)(37,47,48,38)(43,46,45,44);(1,2,6,7)(3,5,9,14)(4,11,13,8)(10,16,15,12)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,39,36)(34,40,42,41)(37,45,48,43)(38,44,47,46);(2,7)(5,14)(8,11)(12,16)(17,19,23,21)(18,25,20,24)(22,27,30,29)(26,32,28,31)(33,36,39,35)(34,41,42,40)(37,45,48,43)(38,44,47,46);(1,3,6,9)(2,5,7,14)(4,10,13,15)(8,12,11,16)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,37)(34,38)(35,43)(36,45)(39,48)(40,46)(41,44)(42,47);(1,4,6,13)(2,8,7,11)(3,10,9,15)(5,12,14,16)(17,21,23,19)(18,24,20,25)(22,29,30,27)(26,31,28,32)(33,37)(34,38)(35,43)(36,45)(39,48)(40,46)(41,44)(42,47);(1,5)(2,9)(3,7)(4,12)(6,14)(8,15)(10,11)(13,16)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,38)(34,37)(35,44)(36,46)(39,47)(40,45)(41,43)(42,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8137897);