Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(10,13)(11,15)(17,18,21,20)(19,22,23,24)(25,26)(27,32)(28,29)(30,31)(33,34)(35,39)(36,40)(37,38);(1,2,5,4)(3,6,7,8)(10,13)(14,16)(19,23)(22,24)(26,31)(27,35)(28,34)(37,40);(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)(25,27,30,35)(26,32,31,39)(28,38,34,36)(29,37,33,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,24,23,22)(25,28)(26,33)(27,36)(29,31)(30,34)(32,37)(35,38)(39,40);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(25,29,30,33)(26,34,31,28)(27,37,35,40)(32,36,39,38)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,c^2=f,d^2=fgh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2701848);