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 | |
4875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,16)(11,19,20,15)(12,21,23,17)(13,18,22,24)(25,26,31,34)(27,32,35,29)(28,30,33,39)(36,37,40,38);(2,3)(6,7)(9,11,14,20)(10,15,16,19)(12,22,23,13)(17,24,21,18)(25,27,31,35)(26,32,34,29)(28,36,33,40)(30,37,39,38);(1,3,5,2)(4,7,8,6)(10,16)(15,19)(17,21)(18,24)(27,35)(29,32)(36,40)(37,38);(1,4)(2,6)(3,7)(5,8)(9,12,14,23)(10,17,16,21)(11,13,20,22)(15,18,19,24)(25,28)(26,30)(27,36)(29,38)(31,33)(32,37)(34,39)(35,40);(9,10)(11,15)(12,17)(13,18)(14,16)(19,20)(21,23)(22,24)(25,29)(26,27)(28,38)(30,36)(31,32)(33,37)(34,35)(39,40);(9,13)(10,18)(11,12)(14,22)(15,17)(16,24)(19,21)(20,23)(25,30)(26,33)(27,37)(28,34)(29,36)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8429339);