Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
299 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 79 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)(4,5)(6,8)(10,13)(11,15)(14,16)(17,18)(19,21,23,24)(20,22);(2,5)(6,7)(9,10,11,13,12,14,15,16)(17,19)(18,21,22,24)(20,23);(1,3,4,8)(2,6,5,7)(10,14)(13,16)(19,23)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,d^2=gh,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #97175, #97176, #97177, … |
| maximal normal subgroup of | #97175, #97176, #97177, … |
| maximal quotient of | #102399, #102402, #102405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2875);