Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2657 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
582 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 582 conjugacy classes of subgroups — too many to draw. Only its 143 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)(9,10,15,21,14,12,13,17)(11,16,22,24,19,18,20,23);(1,2,4,3)(5,6,7,8)(9,11,14,19)(10,16,12,18)(13,20,15,22)(17,23,21,24);(6,8)(9,12,14,10)(11,18,19,16)(13,17,15,21)(20,23,22,24);(1,3,4,2)(11,19)(16,18)(20,22)(23,24)Polycyclic presentation:
2^8|a^2=fh,b^2=egh,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28457, #6465543, #6465553, … |
| maximal normal subgroup of | C2×#28457, #6465543, #6465553, … |
| maximal quotient of | #321781, #321782, #321783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21445);