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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3863 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)(3,12)(4,14)(5,6)(7,9)(8,13)(10,16)(11,15)(34,38)(36,43)(39,44)(42,47);(1,3,7,13)(2,8,9,12)(4,15,16,5)(6,10,11,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(35,41)(39,44)(42,47)(46,48);(2,9)(6,11)(8,12)(10,14)(18,23)(21,31)(24,29)(25,32)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4,7,16)(2,10,9,14)(3,5,13,15)(6,12,11,8)(17,19)(18,24)(20,26)(21,25)(22,28)(23,29)(27,30)(31,32)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,5)(2,11)(3,4)(6,9)(7,15)(8,14)(10,12)(13,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,32)(24,31)(26,28)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,6,7,11)(2,5,9,15)(3,10,13,14)(4,12,16,8)(17,21,22,31)(18,26,23,27)(19,25,28,32)(20,29,30,24)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229235);