Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 143 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 — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,16,19,17,20,25,28)(15,21,22,26,23,27,24,18);(1,2)(3,4)(6,9)(10,12);(2,4)(7,11)(10,12)(13,15,17,23)(14,18,20,26)(16,24,25,22)(19,27,28,21);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #122548, #122572, #122588, … |
| maximal normal subgroup of | #122548, #122572, #122588, … |
| maximal quotient of | #330149, #330153, #330158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22796);