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 | |
4203 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)(3,9)(4,12)(5,11)(6,7)(8,13)(10,15)(14,16)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(2,7)(4,13)(5,9)(10,14)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(1,3)(2,5)(4,10)(6,11)(7,9)(8,15)(12,16)(13,14)(17,19,21,27)(18,22,24,25)(20,26,29,28)(23,30,32,31)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,3)(2,5)(4,14)(6,11)(7,9)(8,16)(10,13)(12,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4)(2,8)(3,10)(5,15)(6,13)(7,12)(9,16)(11,14)(17,20)(18,23)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,5)(2,3)(4,15)(6,9)(7,11)(8,10)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9297697);