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 | |
4672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,18,16,12)(13,21,22,17)(19,20,23,24)(25,26)(27,32)(28,34)(29,39)(30,36)(31,33)(35,37)(38,40);(3,7)(6,8)(10,15)(11,16)(17,21)(19,23)(25,27,31,36)(26,32,33,30)(28,37,38,29)(34,35,40,39);(1,2)(3,6)(4,5)(7,8)(10,15)(12,18)(17,21)(20,24)(26,33)(27,36)(29,37)(34,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,28)(26,34)(27,29)(30,35)(31,38)(32,39)(33,40)(36,37);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,29)(26,35)(27,28)(30,34)(31,37)(32,40)(33,39)(36,38);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,30)(26,27)(28,35)(29,34)(31,32)(33,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8023965);