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 | |
4564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10)(11,15)(12,20)(13,21)(14,16)(17,23)(18,24)(19,22)(25,26,31,29)(27,34,33,30)(28,36,37,32)(35,38,39,40);(1,2,5,8)(3,6,4,7)(9,11,14,19)(10,15,16,22)(12,18,23,21)(13,20,24,17)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,40)(37,39);(1,3,5,4)(2,6,8,7)(10,16)(11,19)(17,20)(18,21)(28,37)(32,36)(35,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,20)(14,19)(16,22)(17,24)(18,23)(25,28)(26,32)(27,35)(29,36)(30,38)(31,37)(33,39)(34,40);(1,4,5,3)(2,7,8,6)(9,12,14,23)(10,17,16,20)(11,18,19,21)(13,22,24,15)(25,29,31,26)(27,34,33,30)(28,32,37,36)(35,38,39,40);(9,13)(10,18)(11,20)(12,22)(14,24)(15,23)(16,21)(17,19)(25,30,31,34)(26,33,29,27)(28,38,37,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064390);