Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = C23×#7684106 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
203 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 89 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:
(1,2)(3,4)(5,7)(6,8)(9,10,13,18,14,19,22,17)(11,12,16,23,21,24,20,15);(1,3,5,8,6,7,4,2)(9,11,13,16,14,21,22,20)(10,15,18,12,19,23,17,24);(1,4,6,5)(2,3,8,7)(9,12,22,15,14,24,13,23)(10,16,17,11,19,20,18,21)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=eg,d^2=g,e^2=h,f^2=g,b^a=bd,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8258, #182977, #182981, … |
| maximal normal subgroup of | C2×#8258, #182977, #182981, … |
| maximal quotient of | #96781, #96783, #96788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1246);