Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C25×C4×D4 (order 1024) · Inn(G) = C2×D4 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C8×D4 |
| prime radical | C23 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
136 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20,14,21,31,38,13,18,30,22,29,19,36,37)(11,12,25,27,24,28,35,40,23,16,32,33,39,26,17,34);(2,5)(6,8)(9,11,15,25,14,24,31,35,13,23,30,32,29,39,36,17)(10,16,20,33,21,26,38,34,18,12,22,27,19,28,37,40);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,17,18,35)(11,22,23,20)(14,28,29,26)(15,27,30,33)(19,32,21,25)(24,37,39,38)(31,40,36,34)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=d,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,e^c=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13106, #494773, #494774, … |
| maximal normal subgroup of | C2×#13106, #494773, #494774, … |
| maximal quotient of | #140066, #140067, #140073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6094);