Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
528 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 135 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 — 4 generators on 20 points:
(5,6,12,15,11,17,10,16)(7,8,9,19,14,20,18,13);(1,2,3,4)(5,7,10,18,11,14,12,9)(6,13,15,8,17,19,16,20);(2,4)(5,8)(6,14)(7,17)(9,15)(10,19)(11,20)(12,13)(16,18);(5,9,10,7,11,18,12,14)(6,8,15,19,17,20,16,13)Polycyclic presentation:
2^8|a^2=fh,b^2=eg,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28500, #6465592, #6465593, … |
| maximal normal subgroup of | C2×#28500, #6465592, #6465593, … |
| maximal quotient of | #321598, #321599, #321601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21488);