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 | |
4532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,7)(5,8)(9,10)(11,12)(13,14)(15,16)(21,33)(26,35)(29,43)(32,44)(37,40)(38,45)(42,48)(46,47);(1,2,4,6)(3,5,7,8)(10,14)(11,15)(18,23)(20,31)(24,27)(26,35)(28,41)(32,44)(37,40)(42,48);(1,3)(2,5)(4,7)(6,8)(9,11,13,15)(10,12,14,16)(17,18)(19,27)(20,25)(21,26)(22,23)(24,30)(28,39)(29,40)(31,34)(32,38)(33,35)(36,41)(37,43)(42,47)(44,45)(46,48);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,19,22,30)(18,24,23,27)(20,28,31,41)(21,29,33,43)(25,36,34,39)(26,37,35,40)(32,42,44,48)(38,46,45,47);(1,3)(2,5)(4,7)(6,8)(17,20,22,31)(18,25,23,34)(19,28,30,41)(21,32,33,44)(24,36,27,39)(26,38,35,45)(29,42,43,48)(37,46,40,47);(17,21)(18,26)(19,29)(20,32)(22,33)(23,35)(24,37)(25,38)(27,40)(28,42)(30,43)(31,44)(34,45)(36,46)(39,47)(41,48)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021551);