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 | |
3718 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)(3,11)(4,15)(5,7)(6,16)(8,14)(9,13)(10,12)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(33,34,39,42)(35,37,40,36)(38,46,47,41)(43,44,48,45);(1,3,7,14)(2,8,5,11)(4,6,13,12)(9,16,15,10)(18,23)(19,24)(25,30)(26,31)(33,35,39,40)(34,36,42,37)(38,43,47,48)(41,44,46,45);(1,2,7,5)(3,8,14,11)(4,15,13,9)(6,10,12,16)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,48,46,43);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,10,11,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,37,39,36)(34,35,42,40)(38,45,47,44)(41,43,46,48);(1,5,7,2)(3,8,14,11)(4,15,13,9)(6,16,12,10)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)(33,38,39,47)(34,41,42,46)(35,43,40,48)(36,44,37,45);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,9,11,15)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=g,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, 8022607);