Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #54982 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,15,12)(11,18,20,16)(13,22,14,17)(19,23,21,24)(25,26)(27,29)(28,36)(30,34)(31,33)(32,38)(35,37)(39,40);(1,3,5,7)(2,4,6,8)(9,11,15,20)(10,16,12,18)(13,19,14,21)(17,23,22,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,37)(36,39)(38,40);(1,4)(2,3)(5,8)(6,7)(9,12,15,10)(11,18,20,16)(13,17,14,22)(19,23,21,24)(25,28,30,38)(26,32,34,36)(27,35,33,40)(29,39,31,37);(3,7)(4,8)(9,13,15,14)(10,17,12,22)(11,19,20,21)(16,23,18,24)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,35,36,40);(2,6)(3,7)(11,20)(16,18)(19,21)(23,24)(26,34)(27,33)(32,36)(35,40);(9,14,15,13)(10,17,12,22)(11,21,20,19)(16,23,18,24)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012728);