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 | |
4597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,17)(12,21)(13,22)(14,16)(15,20)(18,24)(19,23)(25,26,31,36)(27,34,29,32)(28,38,39,33)(30,35,37,40);(2,5)(6,7)(9,11,14,20)(10,15,16,17)(12,18,22,23)(13,24,21,19)(25,27,31,29)(26,32,36,34)(28,30,39,37)(33,35,38,40);(1,2)(3,6)(4,5)(7,8)(10,16)(12,22)(15,17)(18,23)(25,28,31,39)(26,33,36,38)(27,37,29,30)(32,40,34,35);(1,3)(2,6)(4,8)(5,7)(9,12)(10,13)(11,18)(14,22)(15,24)(16,21)(17,19)(20,23)(25,29,31,27)(26,34,36,32)(28,30,39,37)(33,35,38,40);(9,11)(10,15)(12,23)(13,19)(14,20)(16,17)(18,22)(21,24)(25,30)(26,35)(27,28)(29,39)(31,37)(32,33)(34,38)(36,40);(9,13)(10,12)(11,19)(14,21)(15,23)(16,22)(17,18)(20,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=ghi,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, 8758042);