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 | |
4065 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(34,40)(35,37)(42,46)(43,45);(2,8)(4,9)(7,14)(10,15)(18,20)(24,26)(25,28)(31,32)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,22,23,19)(18,26,20,24)(21,27,29,30)(25,31,28,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,46)(41,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=i,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9837451);