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 | |
4579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,15)(11,19,20,16)(12,13,18,17)(21,22,23,24);(3,7)(6,8)(10,15)(11,20)(12,18)(22,24)(25,26,30,31)(27,32,35,39)(28,36,38,29)(33,34,37,40);(1,2,5,4)(3,6,7,8)(10,15)(13,17)(16,19)(22,24)(26,31)(29,36)(32,39)(34,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,23)(13,22)(14,20)(15,19)(17,24)(18,21)(25,27)(26,32)(28,37)(29,34)(30,35)(31,39)(33,38)(36,40);(1,4,5,2)(3,8,7,6)(9,12,14,18)(10,17,15,13)(11,21,20,23)(16,22,19,24)(25,28)(26,29)(27,33)(30,38)(31,36)(32,40)(34,39)(35,37);(9,13,14,17)(10,18,15,12)(11,22,20,24)(16,21,19,23)(25,29)(26,28)(27,34)(30,36)(31,38)(32,37)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8258860);