Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 32768 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
607 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 607 conjugacy classes of subgroups — too many to draw. Only its 123 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)(9,10,15,19,14,11,13,17)(12,16,22,24,21,18,20,23);(5,6)(7,8)(9,11,14,10)(12,18,21,16)(13,17,15,19)(20,23,22,24);(2,4)(6,8)(9,12,14,21)(10,16,11,18)(13,20,15,22)(17,23,19,24);(2,4)(9,13)(10,11)(12,20)(14,15)(16,18)(21,22)Polycyclic presentation:
2^8|a^2=efh,b^2=h,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125646, #125886, #126284, … |
| maximal normal subgroup of | #125646, #125886, #126284, … |
| maximal quotient of | #343069, #343070, #343071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24056);