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 | C2×Q8×#121 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(3,4,5,6)(7,10,9,8);(3,7,5,9)(4,8,6,10);(12,14)(16,18);(15,16,17,18);(11,12)(13,14)140 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7684310, C2×#41×#121, C2×#7689762, … |
| maximal normal subgroup of | C2×#7684310, C2×#41×#121, C2×#7689762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537439);