Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C23×D4×#392 (order 4096) · Inn(G) = C2×D4 · Out(G) = C22×#392 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
234 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 95 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:
(1,2,5,3,4,6,8,7)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(2,6)(3,7)(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24);(1,3,5,6,4,7,8,2)(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,b^a=bd,c^a=ch,d^a=dg,f^a=fg,c^b=cgh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8197, #182349, #182350, … |
| maximal normal subgroup of | C2×#8197, #182349, #182350, … |
| maximal quotient of | #96702, #96703, #96706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1185);