Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#9674 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(8,11,13,14)(15,16,17,20,18,21,22,19);(7,8,12,11,10,13,9,14)(16,19)(17,22)(20,21);(3,4,5,6)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#210434, C2×#231412, C2×D4×D4⋊C8, … |
| maximal normal subgroup of | C2×#210434, C2×#231412, C2×D4×D4⋊C8, … |
| maximal quotient of | C2×#140846 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111679);