Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,19,22,24)(18,23);(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24);(2,5)(6,8)(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=e,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,e^b=ef,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94168, #94170, #94172, … |
| maximal normal subgroup of | #94168, #94170, #94172, … |
| maximal quotient of | #124657, #124687, #124714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4490);