Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
232 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 61 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)(3,7,8,5)(4,6)(9,10)(11,19)(12,15)(13,23)(14,18)(16,21)(17,22)(20,24);(1,3)(2,5)(4,8)(6,7)(10,15)(11,20)(12,21)(13,22)(16,17)(18,24)(19,23);(3,8)(5,7)(9,11,13,12,14,21,22,20)(10,15,17,19,18,24,23,16)Polycyclic presentation:
2^8|a^2=e,c^2=g,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,c^b=cfh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130244, #130254, #130726, … |
| maximal normal subgroup of | #130244, #130254, #130726, … |
| maximal quotient of | #122704, #122709, #122713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4876);