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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,13,14,9)(5,10,15,16)(33,34,39,44)(35,45,47,40)(36,41,46,42)(37,43,48,38);(2,7)(3,12)(9,13)(10,16)(17,18,22,23)(19,24,26,20)(21,25,31,32)(27,30,29,28)(33,35)(34,40)(36,43)(37,41)(38,46)(39,47)(42,48)(44,45);(2,7)(3,12)(9,13)(10,16)(18,23)(19,26)(25,32)(27,29)(33,36,39,46)(34,41,44,42)(35,43,47,38)(37,45,48,40);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,9,15,13)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,31)(30,32)(33,36)(34,42)(35,43)(37,45)(38,47)(39,46)(40,48)(41,44);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,8,15,11)(17,20,22,24)(18,19,23,26)(21,30,31,28)(25,29,32,27)(33,37,39,48)(34,43,44,38)(35,41,47,42)(36,45,46,40);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,21,22,31)(18,25,23,32)(19,27,26,29)(20,28,24,30)(33,38,39,43)(34,37,44,48)(35,46,47,36)(40,41,45,42)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8633463);