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 | |
3821 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18)(19,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,34,39,37)(35,43,44,40)(36,42,38,41)(45,47,46,48);(2,7)(5,8)(9,12)(13,15)(17,18,23,20)(19,24,21,27)(22,28,30,25)(26,32,29,31)(34,37)(35,44)(41,42)(45,46);(2,7)(3,10)(9,12)(11,16)(17,19,23,21)(18,24,20,27)(22,26,30,29)(25,31,28,32)(33,35,39,44)(34,40,37,43)(36,45,38,46)(41,47,42,48);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,20,23,18)(19,24,21,27)(22,28,30,25)(26,31,29,32)(33,36,39,38)(34,41,37,42)(35,45,44,46)(40,47,43,48);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,21,23,19)(18,24,20,27)(22,29,30,26)(25,31,28,32)(33,37,39,34)(35,43,44,40)(36,42,38,41)(45,48,46,47);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,22)(18,25)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32)(33,38,39,36)(34,42,37,41)(35,46,44,45)(40,48,43,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=ghi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664610);