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 | |
3935 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)(19,22)(20,23)(21,24)(25,26)(27,32)(28,31)(29,30)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(2,7)(4,8)(10,13)(12,14)(17,19)(18,22)(20,26)(21,29)(23,25)(24,30)(27,31)(28,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,7)(3,9)(10,13)(11,16)(18,20)(19,25)(24,28)(27,29)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,14)(12,13)(15,16)(17,20,23,18)(19,26,25,22)(21,24,31,28)(27,32,29,30)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,21)(18,24)(19,27)(20,28)(22,32)(23,31)(25,29)(26,30)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,22,23,26)(18,25,20,19)(21,30,31,32)(24,27,28,29)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113803);