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 | |
4880 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)(4,7)(6,8)(10,12)(14,16)(17,18,23,26)(19,27,28,20)(21,33,34,24)(22,36,37,25)(29,31,41,39)(30,32,42,40)(35,38,46,45)(43,47,48,44);(2,3)(6,7)(11,15)(14,16)(19,28)(20,27)(21,34)(24,33)(30,42)(32,40)(35,46)(38,45);(1,2,5,3)(4,6,8,7)(9,10,13,12)(11,14,15,16)(17,19,23,28)(18,20,26,27)(21,29,34,41)(22,30,37,42)(24,31,33,39)(25,32,36,40)(35,43,46,48)(38,44,45,47);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,31)(22,32)(23,27)(24,29)(25,30)(26,28)(33,41)(34,39)(35,44)(36,42)(37,40)(38,43)(45,48)(46,47);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,21,23,34)(18,24,26,33)(19,29,28,41)(20,31,27,39)(22,35,37,46)(25,38,36,45)(30,43,42,48)(32,44,40,47);(17,22,23,37)(18,25,26,36)(19,30,28,42)(20,32,27,40)(21,35,34,46)(24,38,33,45)(29,43,41,48)(31,44,39,47)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022658);