Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
535 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 535 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,4)(9,10,12,15,13,11,16,14);(1,2,3,4)(5,6)(7,8);(6,8)(9,11)(10,13)(12,14)(15,16)Polycyclic presentation:
2^8|a^2=g,b^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cf,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94277, #94278, #94281, … |
| maximal normal subgroup of | #94277, #94278, #94281, … |
| maximal quotient of | #115703, #115704, #115705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4041);