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 | |
3943 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)(17,18)(19,24)(20,28)(21,25)(22,27)(23,29)(26,30)(31,32)(33,34,40,45)(35,41,47,48)(36,44,38,42)(37,43,46,39);(2,7)(3,12)(9,13)(10,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35,40,47)(34,41,45,48)(36,37,38,46)(39,44,43,42);(2,7)(3,12)(9,13)(10,16)(20,29)(23,28)(26,32)(30,31)(33,36,40,38)(34,42,45,44)(35,46,47,37)(39,41,43,48);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,9,15,13)(17,20,21,29)(18,23,25,28)(19,26,27,32)(22,30,24,31)(33,37,40,46)(34,43,45,39)(35,36,47,38)(41,44,48,42);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,8,15,11)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,30)(29,32)(33,38,40,36)(34,44,45,42)(35,46,47,37)(39,41,43,48);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,18)(19,22)(20,28)(21,25)(23,29)(24,27)(26,31)(30,32)(33,39,40,43)(34,37,45,46)(35,44,47,42)(36,41,38,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=gi,e^2=h,f^2=h,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, 8627640);