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)(3,9)(4,12)(5,10)(6,7)(8,13)(11,15)(14,16)(17,18)(19,26)(20,21)(22,23)(24,27)(25,31)(28,29)(30,32)(33,34,38,41)(35,36,39,43)(37,40,46,48)(42,44,45,47);(2,7)(4,13)(5,9)(11,14)(18,23)(21,25)(24,26)(29,32)(33,35)(34,36)(37,45)(38,39)(40,47)(41,43)(42,46)(44,48);(2,7)(3,10)(8,12)(11,14)(19,27)(24,26)(28,30)(29,32)(35,39)(36,43)(42,45)(44,47);(1,3)(2,5)(4,14)(6,10)(7,9)(8,16)(11,13)(12,15)(17,19)(18,24)(20,30)(21,29)(22,27)(23,26)(25,32)(28,31)(33,36,38,43)(34,39,41,35)(37,44,46,47)(40,45,48,42);(1,4)(2,8)(3,11)(5,15)(6,13)(7,12)(9,16)(10,14)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,32)(27,30)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,47)(46,48);(1,5)(2,3)(4,15)(6,9)(7,10)(8,11)(12,14)(13,16)(17,21,22,25)(18,20,23,31)(19,29,27,32)(24,30,26,28)(33,37)(34,40)(35,42)(36,44)(38,46)(39,45)(41,48)(43,47)Polycyclic presentation:
2^9|a^2=i,d^2=i,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=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8747493);