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 | |
3776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,8,10,5)(4,9,13,14)(11,15,16,12)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,3,7,10)(2,8,6,5)(4,11,13,16)(9,15,14,12)(18,22)(19,26)(24,28)(27,32)(34,39)(35,40)(41,46)(42,47);(3,10)(5,8)(11,16)(12,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,4)(2,9)(3,11)(5,12)(6,14)(7,13)(8,15)(10,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,5)(2,3)(4,12)(6,10)(7,8)(9,11)(13,15)(14,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48);(1,6,7,2)(3,5,10,8)(4,9,13,14)(11,15,16,12)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8458220);