Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 369 conjugacy classes of subgroups — too many to draw. Only its 163 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18,15,20,13,19)(11,21,12,17,23,24,22,16);(2,5)(6,8)(9,11,14,12,15,23,13,22)(10,16,19,24,20,17,18,21);(1,3,4,7)(2,6,5,8)(9,11,13,22,15,23,14,12)(10,17,18,24,20,16,19,21);(9,12,14,23,15,22,13,11)(10,17,18,24,20,16,19,21)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=e,d^2=g,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124409, #128893, #134565, … |
| maximal normal subgroup of | #124409, #128893, #134565, … |
| maximal quotient of | #212712, #212713, #212714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14650);