Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2212 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C7×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C4×C7×C8 ◁ G | |
154 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18)(11,23)(12,27)(13,29)(14,19)(15,21)(16,20)(17,28)(22,36)(24,35)(25,39)(26,31)(30,32)(33,37)(34,38);(8,10,12,21,14,22,28,13)(9,15,17,18,19,29,27,36)(11,20,24,37,26,38,25,30)(16,31,32,35,34,23,33,39);(8,11)(9,16)(10,20)(12,24)(13,30)(14,26)(15,31)(17,32)(18,35)(19,34)(21,37)(22,38)(23,29)(25,28)(27,33)(36,39)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5447);