Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
539 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 207 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10,13,16)(11,14,18,23)(12,15,20,17)(19,21,24,22);(1,2,4,3)(5,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24);(1,3,4,2)(10,15)(12,20)(14,21)(16,17)(19,24)(22,23);(10,16)(11,18)(15,17)(19,24)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=e,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #360281, #361001, #368501, … |
| maximal normal subgroup of | #360281, #361001, #368501, … |
| maximal quotient of | #202328, #202343, #202358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14054);