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 | |
3773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,24)(20,21)(22,32)(23,26)(25,30)(27,31)(28,29)(33,34,39,37)(35,43,44,40)(36,42,38,41)(45,47,46,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,23,22)(18,24,26,32)(20,27,30,29)(21,31,25,28)(34,37)(35,44)(41,42)(45,46);(2,6)(7,10)(8,12)(14,16)(17,20)(18,25)(19,27)(21,26)(22,29)(23,30)(24,28)(31,32)(33,35,39,44)(34,40,37,43)(36,45,38,46)(41,47,42,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(33,36,39,38)(34,41,37,42)(35,45,44,46)(40,47,43,48);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,21,23,25)(18,20,26,30)(19,28,22,31)(24,29,32,27)(33,37,39,34)(35,43,44,40)(36,42,38,41)(45,48,46,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,19)(18,24,26,32)(20,29,30,27)(21,31,25,28)(33,38,39,36)(34,42,37,41)(35,46,44,45)(40,48,43,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,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, 8663274);