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 | |
4254 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,40)(36,42)(37,38)(39,41)(43,47)(44,48)(45,46);(2,6)(4,13)(7,9)(11,16)(18,21)(23,25)(24,29)(30,31)(33,35)(34,40)(36,43)(37,39)(38,41)(42,47)(44,45)(46,48);(2,6)(3,10)(8,12)(11,16)(20,28)(24,29)(26,32)(30,31)(34,41)(35,37)(36,44)(46,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,32)(29,31)(33,36,39,44)(34,42,41,48)(35,43,37,45)(38,46,40,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37,39,35)(34,38,41,40)(36,45,44,43)(42,46,48,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,18)(19,25,27,23)(20,29,28,24)(26,31,32,30)(33,38)(34,35)(36,46)(37,41)(39,40)(42,43)(44,47)(45,48)Polycyclic presentation:
2^9|a^2=ghi,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9197627);