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 | |
3981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,10,5)(13,16,15,14)(17,18,22,26)(19,23,27,31)(20,24,30,28)(21,25,32,29)(33,34)(35,44)(36,42)(37,41)(38,40)(39,43)(45,47)(46,48);(2,7)(3,12)(4,10)(14,16)(17,19)(18,23)(20,29)(21,24)(22,27)(25,30)(26,31)(28,32)(33,35,39,38)(34,40,43,44)(36,45,37,46)(41,47,42,48);(3,12)(8,11)(13,15)(14,16)(19,27)(20,30)(23,31)(24,28)(33,36,39,37)(34,41,43,42)(35,45,38,46)(40,47,44,48);(1,3)(2,8)(4,15)(5,14)(6,12)(7,11)(9,16)(10,13)(17,20)(18,24)(19,25)(21,31)(22,30)(23,32)(26,28)(27,29)(33,37,39,36)(34,42,43,41)(35,45,38,46)(40,47,44,48);(1,4)(2,9)(3,13)(5,7)(6,10)(8,14)(11,16)(12,15)(17,21)(18,25)(19,28)(20,23)(22,32)(24,27)(26,29)(30,31)(33,38,39,35)(34,40,43,44)(36,46,37,45)(41,47,42,48);(1,5,6,9)(2,10,7,4)(3,14,12,16)(8,13,11,15)(17,18)(19,23)(20,24)(21,29)(22,26)(25,32)(27,31)(28,30)(33,34)(35,40)(36,41)(37,42)(38,44)(39,43)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,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, 10219565);