Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
769 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 769 conjugacy classes of subgroups — too many to draw. Only its 127 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 — 4 generators on 16 points:
(1,2,3,4)(9,10,11,15,12,13,16,14);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16);(2,4)(6,8)(10,13)(14,15);(2,4)(10,14)(11,16)(13,15)Polycyclic presentation:
2^8|a^2=efh,e^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #123091, #123101, #125644, … |
| maximal normal subgroup of | #123091, #123101, #125644, … |
| maximal quotient of | #343055, #343056, #343057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24052);