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 | |
3917 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,12)(3,8,14,15)(4,11,6,9)(5,10,16,13)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3,7,14)(2,8,12,15)(4,10,6,13)(5,11,16,9)(18,23)(19,26)(25,28)(27,32)(34,38)(39,41)(40,45)(46,47);(1,4,7,6)(2,9,12,11)(3,13,14,10)(5,8,16,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5,7,16)(2,10,12,13)(3,9,14,11)(4,15,6,8)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,6,7,4)(2,11,12,9)(3,13,14,10)(5,8,16,15)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,2)(3,8)(4,9)(5,10)(6,11)(7,12)(13,16)(14,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014790);