Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 16384 |
| 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 | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C23 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
335 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 187 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)(3,8)(4,6)(5,7)(9,10,14,18,15,19,13,23)(11,20,12,17,21,24,22,16);(1,3,5,4)(2,6,7,8)(11,21)(12,22)(16,17)(20,24);(9,11,13,22,15,21,14,12)(10,16,18,20,19,17,23,24);(1,4,5,3)(2,6,7,8)(9,12,14,21,15,22,13,11)(10,17,18,24,19,16,23,20)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=eg,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124802, #124816, #124818, … |
| maximal normal subgroup of | #124802, #124816, #124818, … |
| maximal quotient of | #213407, #213410, #213412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14724);