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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3711 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)(8,15)(10,13)(11,16)(17,18,23,27)(19,21,25,32)(20,26,22,24)(28,31,29,30)(34,40)(35,45)(36,38)(43,48);(1,2,6,7)(3,8,9,15)(4,12,14,5)(10,11,13,16)(17,19)(18,21)(20,28)(22,29)(23,25)(24,30)(26,31)(27,32)(33,34)(35,46)(36,42)(37,48)(38,44)(39,40)(41,45)(43,47);(2,7)(5,12)(8,15)(11,16)(17,20,23,22)(18,24,27,26)(19,28,25,29)(21,30,32,31)(33,35)(34,41)(36,37)(38,47)(39,45)(40,46)(42,43)(44,48);(1,3)(2,8)(4,13)(5,11)(6,9)(7,15)(10,14)(12,16)(33,36,39,38)(34,42,40,44)(35,37,45,47)(41,43,46,48);(1,4)(2,5)(3,10)(6,14)(7,12)(8,16)(9,13)(11,15)(17,21)(18,25)(19,27)(20,30)(22,31)(23,32)(24,29)(26,28)(33,37,39,47)(34,43,40,48)(35,36,45,38)(41,42,46,44);(1,5)(2,4)(3,11)(6,12)(7,14)(8,13)(9,16)(10,15)(17,22,23,20)(18,26,27,24)(19,29,25,28)(21,31,32,30)(33,38,39,36)(34,44,40,42)(35,47,45,37)(41,48,46,43)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833672);