Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×C4×#353 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C23 | C2×#99 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(3,4,5,6);(7,8)(9,12,14,11)(10,13);(8,11)(9,14)(12,13)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#2196, C4×#12759, C4×#12693, … |
| maximal normal subgroup of | C2×C4×#2196, C4×#12759, C4×#12693, … |
| maximal quotient of | C4×#12693, #72282658, C4×#12797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400197);