Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
| direct product | D4×#392 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
13174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,3,4);(2,4);(10,12)(13,14)(15,16);(6,8)(10,12)(14,16);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)110 conjugacy classes — view the full character table.
| maximal subgroup of | #72213999, #72215608, C2×D4×#392, … |
| maximal normal subgroup of | #72213999, #72215608, C2×D4×#392, … |
| maximal quotient of | C2×D4×#392, #72233926, #72233927, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213553);