Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
329 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 153 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)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(2,5)(3,8)(6,7)(11,17)(14,20)(18,23)(21,24);(2,6)(5,7)(9,11,12,18,13,17,19,23)(10,14,16,21,15,20,22,24);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^8|a^2=f,c^2=f,e^2=g,f^2=h,b^a=be,c^a=cg,d^a=dh,e^a=eg,c^b=ch,e^b=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | #137857, #209092, #209094, … |
| maximal normal subgroup of | #137857, #209092, #209094, … |
| maximal quotient of | #182513, #182550, #183468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10921);