Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
521 conjugacy classes of subgroups; only the 343 normal subgroups are shown.
This group has 521 conjugacy classes of subgroups — too many to draw. Only its 343 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,10,12,15)(11,17,18,20);(1,2,3,4)(5,6)(7,11)(8,15)(9,12)(10,16)(13,18)(14,20)(17,19);(5,7,8,14,9,13,16,19)(6,11,15,20,12,18,10,17);(7,13)(11,18)(14,19)(17,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,e^2=h,b^a=be,e^b=eh,f^b=fh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #126894, #126895, #126896, … |
| maximal normal subgroup of | #126894, #126895, #126896, … |
| maximal quotient of | #183337, #183340, #183345, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13433);