Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(10,13)(11,15)(17,18,21,20)(19,22,23,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,2,5,3)(4,7,8,6)(11,15)(14,16)(19,23)(22,24)(26,31)(27,32)(33,40)(34,37);(2,3)(6,7)(9,10,12,13)(11,14,15,16)(17,19,21,23)(18,22,20,24)(25,27,30,32)(26,28,31,36)(29,37,39,34)(33,38,40,35);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,24,23,22)(25,28)(26,32)(27,31)(29,38)(30,36)(33,34)(35,39)(37,40);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(25,29,30,39)(26,33,31,40)(27,34,32,37)(28,35,36,38)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=gh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4079156);