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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
265 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,9,10,11)(13,14,17,21,18,22,16,20)(15,23,24,27,26,28,25,19);(5,6,7,11,8,10,12,9)(13,15,18,26)(14,19)(16,25,17,24)(20,28)(21,23)(22,27);(2,4)(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,23,27,28)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=g,d^2=gh,f^2=g,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 | #98062, #98063, #98064, … |
| maximal normal subgroup of | #98062, #98063, #98064, … |
| maximal quotient of | #99744, #99746, #99748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2114);