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 | |
3782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,10,8)(4,9,14,16)(11,15,12,13)(33,34,40,44)(35,41,39,37)(36,45,47,42)(38,43,48,46);(2,7)(3,10)(9,16)(11,12)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(33,35,40,39)(34,41,44,37)(36,46,47,43)(38,42,48,45);(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,36,40,47)(34,42,44,45)(35,43,39,46)(37,48,41,38);(1,3,6,10)(2,8,7,5)(4,12,14,11)(9,15,16,13)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,37,40,41)(34,35,44,39)(36,38,47,48)(42,46,45,43);(1,4,6,14)(2,9,7,16)(3,11,10,12)(5,15,8,13)(17,19)(18,20)(21,29)(22,27)(23,25)(24,31)(26,30)(28,32)(33,38)(34,43)(35,45)(36,37)(39,42)(40,48)(41,47)(44,46);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,11,16,12)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,39,40,35)(34,41,44,37)(36,46,47,43)(38,45,48,42)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154764);