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 | |
3750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,5,9,7)(4,12,14,8)(11,15,16,13)(17,18)(19,27)(20,23)(21,25)(22,30)(24,28)(26,31)(29,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,13)(14,16)(17,19,23,28)(18,24,20,27)(21,31,32,22)(25,30,29,26)(35,42)(39,46)(43,47)(44,48);(18,20)(24,27)(25,29)(26,30)(33,35,38,42)(34,39,41,46)(36,43,40,47)(37,44,45,48);(1,4,6,14)(2,8,10,12)(3,11,9,16)(5,13,7,15)(17,20,23,18)(19,27,28,24)(21,29,32,25)(22,30,31,26);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,15)(14,16)(17,21,23,32)(18,25,20,29)(19,22,28,31)(24,26,27,30)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5)(2,9)(3,10)(4,13)(6,7)(8,16)(11,12)(14,15)(17,22)(18,26)(19,21)(20,30)(23,31)(24,25)(27,29)(28,32)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=i,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592741);