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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,24)(20,29)(21,26)(22,30)(23,25)(27,32)(28,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,23,27)(18,24,25,32)(20,21,30,31)(22,28,29,26)(35,41)(38,45)(42,47)(46,48);(2,9)(3,4)(10,16)(11,12)(18,25)(19,27)(20,30)(26,28)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20)(18,22)(19,21)(23,30)(24,28)(25,29)(26,32)(27,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,21)(18,26)(19,20)(22,32)(23,31)(24,29)(25,28)(27,30)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,22)(18,20)(19,28)(21,24)(23,29)(25,30)(26,27)(31,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,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, 9637826);