Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C82 · Schur multiplier (H2) = C24×C8 |
| direct product | C2×C8×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C82 | C22 |
| commutator | C2 | C22×C82 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×C4×C8 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,5,6,7,8,9,10);(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26);(12,20)(14,22)(16,24)(18,26)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#92909, C2×C8×#347, C2×C8×#509, … |
| maximal normal subgroup of | C2×#92909, C2×C8×#347, C2×C8×#509, … |
| maximal quotient of | C2×#128204, C2×C8×#346, C2×#122211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387091);