Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
547 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 547 conjugacy classes of subgroups — too many to draw. Only its 139 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:
(1,2)(3,4)(9,10)(11,16)(12,21)(13,23)(14,18)(15,19)(17,22)(20,24);(5,6,7,8)(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24);(2,4)(6,8);(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)Polycyclic presentation:
2^8|b^2=eg,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #143703, #143757, #145143, … |
| maximal normal subgroup of | #143703, #143757, #145143, … |
| maximal quotient of | #318005, #318011, #318015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21213);