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 | |
4127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18,23,25)(19,24,27,21)(20,30,26,22)(28,29,31,32)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(4,11)(8,14)(10,15)(12,16)(17,19,23,27)(18,24,25,21)(20,31,26,28)(22,29,30,32)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(2,7)(3,9)(8,14)(10,15)(18,25)(19,27)(20,26)(29,32)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,20,23,26)(18,22,25,30)(19,28,27,31)(21,29,24,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,19)(20,29)(22,28)(23,24)(25,27)(26,32)(30,31)(33,37,39,35)(34,40,36,42)(38,43,46,45)(41,48,44,47);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,22)(18,26)(19,29)(20,25)(21,31)(23,30)(24,28)(27,32)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854150);