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 | |
4285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,14,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(33,34,39,43)(35,44,37,40)(36,41,42,38)(45,48,47,46);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,34)(35,40)(36,41)(37,44)(38,42)(39,43)(45,46)(47,48);(3,11)(7,8)(12,14)(13,16)(20,28)(24,31)(27,29)(30,32)(33,35,39,37)(34,40,43,44)(36,47,42,45)(38,46,41,48);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,36,39,42)(34,41,43,38)(35,45,37,47)(40,46,44,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,21,22,19)(18,23,25,26)(20,27,28,29)(24,32,31,30)(33,37,39,35)(34,40,43,44)(36,45,42,47)(38,46,41,48);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,38,39,41)(34,42,43,36)(35,46,37,48)(40,45,44,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=h,d^2=h,e^2=g,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, 9080323);