Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 425 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,13,15,19)(8,11)(9,12)(10,14,17,20)(16,18);(5,7)(6,10,12,17)(8,14)(9,15)(11,13,18,19)(16,20);(1,2,3,4)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=e,d^2=h,b^a=bd,c^a=ce,g^a=gh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105688, #105689, #106004, … |
| maximal normal subgroup of | #105688, #105689, #106004, … |
| maximal quotient of | #108767, #108768, #108769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3824);