Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | #41×#121 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2)(3,6)(4,5)(7,8);(2,5)(6,8);(1,3,4,7)(2,6,5,8);(10,12)(14,16);(13,14,15,16);(9,10)(11,12)140 conjugacy classes — view the full character table.
| maximal subgroup of | D42×#41, #72257891, #72262524, … |
| maximal normal subgroup of | D42×#41, #72257891, #72262524, … |
| maximal quotient of | #72257264, #72257294, #72257316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537525);