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 | |
3925 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,14)(10,16)(13,15)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30)(33,34,39,40)(35,36,46,41)(37,43,38,42)(44,48,45,47);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(34,40)(36,41)(42,43)(47,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,35,39,46)(34,41,40,36)(37,44,38,45)(42,48,43,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,36,39,41)(34,35,40,46)(37,47,38,48)(42,44,43,45);(33,37,39,38)(34,42,40,43)(35,44,46,45)(36,47,41,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,38,39,37)(34,43,40,42)(35,45,46,44)(36,48,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=h,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,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015246);