Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1914 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)(9,10)(11,14)(12,16)(13,15)(18,21)(22,24)(25,26,29,30)(27,31,33,34)(28,36,38,32)(35,40,37,39);(1,2,5,3)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(19,23)(22,24)(26,30)(27,33)(32,36)(35,37);(2,3)(6,7)(10,15)(11,16)(17,18,20,21)(19,22,23,24)(25,27)(26,31)(28,37)(29,33)(30,34)(32,40)(35,38)(36,39);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,38)(37,39);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,19,20,23)(18,22,21,24)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fgi,c^2=fgh,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5025369);