Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
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 | |
2326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,12,15,14)(17,18)(19,24)(20,21)(22,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,16)(13,15)(18,23)(19,20)(25,26,28,30)(27,31,32,29);(2,6)(5,7)(11,15)(12,14)(17,19,22,20)(18,24,23,21)(25,27,28,32)(26,29,30,31);(1,3)(2,7)(4,8)(5,6)(9,10)(11,12)(13,16)(14,15)(17,20,22,19)(18,21,23,24);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,16)(13,14)(17,21)(18,20)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=gi,b^2=fgh,c^2=hi,d^2=fgi,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.
| maximal subgroup of | #72267449 |
| maximal normal subgroup of | #72267449 |
| maximal quotient of | #72267462 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3456907);