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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4078 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)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34,38,42)(35,41,45,39)(36,37,46,40)(43,44,48,47);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(18,21)(23,25)(24,28)(30,31)(33,35,38,45)(34,39,42,41)(36,43,46,48)(37,47,40,44);(2,6)(7,10)(8,12)(14,16)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(33,36,38,46)(34,37,42,40)(35,43,45,48)(39,47,41,44);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36)(34,40)(35,43)(37,42)(38,46)(39,44)(41,47)(45,48);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,40)(34,36,42,46)(35,44,45,47)(39,48,41,43);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,35)(34,41)(36,43)(37,44)(38,45)(39,42)(40,47)(46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^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, 8663286);