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 | |
4471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,13)(14,16)(18,23)(20,31)(21,33)(24,28)(29,39)(30,40)(38,45)(46,48);(2,5)(6,7)(10,13)(14,15)(18,23)(19,27)(20,31)(26,35)(30,40)(32,44)(36,41)(46,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,28)(20,25)(21,26)(22,23)(24,27)(29,41)(30,42)(31,34)(32,38)(33,35)(36,39)(37,40)(43,48)(44,45)(46,47);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,14,13,15)(17,19,22,27)(18,24,23,28)(20,29,31,39)(21,30,33,40)(25,36,34,41)(26,37,35,42)(32,43,44,47)(38,46,45,48);(1,3,4,8)(2,6,5,7)(17,20,22,31)(18,25,23,34)(19,29,27,39)(21,32,33,44)(24,36,28,41)(26,38,35,45)(30,43,40,47)(37,46,42,48);(17,21,22,33)(18,26,23,35)(19,30,27,40)(20,32,31,44)(24,37,28,42)(25,38,34,45)(29,43,39,47)(36,46,41,48)Polycyclic presentation:
2^9|a^2=ghi,d^2=gh,e^2=h,f^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, 8022691);