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 | |
3989 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18)(19,26)(20,23)(21,25)(22,24)(27,32)(28,29)(30,31)(33,34,39,42)(35,38,41,36)(37,40,46,48)(43,47,45,44);(1,2)(3,8)(4,9)(5,10)(6,7)(11,14)(12,16)(13,15)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35,39,41)(34,36,42,38)(37,45,46,43)(40,47,48,44);(2,7)(3,9)(10,13)(11,16)(18,20)(24,26)(25,28)(31,32)(33,36,39,38)(34,35,42,41)(37,44,46,47)(40,43,48,45);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37)(34,40)(35,43)(36,44)(38,47)(39,46)(41,45)(42,48);(1,5)(2,10)(3,11)(4,12)(6,15)(7,13)(8,14)(9,16)(17,22,23,19)(18,26,20,24)(21,27,29,30)(25,31,28,32)(33,38,39,36)(34,41,42,35)(37,44,46,47)(40,43,48,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9758245);