Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(9,10,12,14)(11,15,16,13)(20,24)(22,23)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,38)(39,40);(2,3)(6,7)(11,16)(13,15)(17,18,21,19)(20,22,24,23)(26,31)(28,35)(33,40)(36,38);(1,2,5,3)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,23,24,22)(25,27,30,32)(26,28,31,35)(29,37,39,34)(33,38,40,36);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,21,24)(18,22,19,23)(25,28)(26,32)(27,31)(29,38)(30,35)(33,34)(36,39)(37,40);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,14)(15,16)(25,29,30,39)(26,33,31,40)(27,34,32,37)(28,36,35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=fgi,c^2=ghi,d^2=fghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3486051);