Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
210 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 73 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)(9,10,15,20)(11,21,12,23)(13,19,14,18)(16,24,17,22);(1,3,5,6,4,7,8,2)(9,11,15,12)(10,16,20,17)(13,24,14,22)(18,23,19,21);(2,6)(3,7)(9,12,15,11)(10,17,20,16)(13,24,14,22)(18,23,19,21)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=dg,e^c=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8842, #191813, #191814, … |
| maximal normal subgroup of | C2×#8842, #191813, #191814, … |
| maximal quotient of | #99332, #99333, #99335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1830);