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 | #52963 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3909 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)(3,6)(4,9)(5,10)(7,11)(8,13)(12,14)(15,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(2,3)(4,11)(8,10)(12,16)(18,22)(23,25)(24,28)(30,31)(34,38)(35,42)(40,44)(43,48);(2,3)(7,9)(8,10)(14,15)(19,26)(23,25)(27,32)(30,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^9|a^2=i,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, 8022632);