Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 67 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 — 3 generators on 24 points:
(2,6,7,8)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22);(1,2)(3,6)(4,8)(5,7)(9,10,13,16)(11,18,20,24)(12,21,19,15)(14,22,23,17);(1,3,5,4)(2,6,7,8)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=fgh,d^2=gh,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #129000, #129002, #129018, … |
| maximal normal subgroup of | #129000, #129002, #129018, … |
| maximal quotient of | #124107, #124219, #124232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4731);