Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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)(3,5)(4,6)(7,8)(10,11)(14,15)(17,18,20,21)(19,22,24,23);(1,3,4,7)(2,5,6,8)(9,10,13,11)(12,14,16,15)(17,19,20,24)(18,22,21,23);(2,6)(5,8)(9,11,13,10)(12,14,16,15)(18,21)(19,23)(22,24);(9,12,13,16)(10,14,11,15)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=e,d^2=e,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=de,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #274401, #274402, #274417, … |
| maximal normal subgroup of | #274401, #274402, #274417, … |
| maximal quotient of | #380026, #380196, #380197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22929);