Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
639 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 639 conjugacy classes of subgroups — too many to draw. Only its 263 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24);(12,20)(15,22)(18,24)(21,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,f^2=h,b^a=be,e^a=eg,f^b=fg,d^c=dg82 conjugacy classes — view the full character table.
| maximal subgroup of | #137684, #137686, #137734, … |
| maximal normal subgroup of | #137684, #137686, #137734, … |
| maximal quotient of | #160505, #160509, #160513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10339);