Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
577 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 577 conjugacy classes of subgroups — too many to draw. Only its 243 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)(3,7)(4,5)(6,8)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23);(2,5)(6,7)(9,11,12,20,13,19,21,18)(10,14,16,24,15,22,23,17);(1,3,4,8)(2,6,5,7);(10,15)(14,22)(16,23)(17,24)Polycyclic presentation:
2^8|a^2=g,b^2=efg,c^2=f,e^2=h,b^a=be,c^a=cf,d^a=dh,e^b=eh,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #107413, #107415, #107505, … |
| maximal normal subgroup of | #107413, #107415, #107505, … |
| maximal quotient of | #206708, #206709, #206710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14294);