Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10336 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator, Frattini, cosocle | C25 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #2403 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(6,12)(13,16)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,32)(28,31);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20,21,27)(18,23,24,31)(19,25,26,32)(22,28,29,30);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)Polycyclic presentation:
2^9|c^2=i,d^2=h,b^a=be,c^a=cf,g^a=ghi,c^b=cg,f^b=fi,e^c=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238035, #72251151, #72251153, … |
| maximal normal subgroup of | #72238035, #72251151, #72251153, … |
| maximal quotient of | #72257816, #72264154, #72264162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140736);