Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C23×C8 |
| socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
393 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 215 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,11,9,7,10,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,28)(25,27);(5,7,8,12,9,6,10,11)(13,15)(14,19)(16,25)(17,23)(18,24)(20,28)(21,22)(26,27);(1,2,3,4)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,22,28);(13,17,18,16)(14,20,21,26)(15,23,24,25)(19,28,22,27)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #99932, #99936, #99940, … |
| maximal normal subgroup of | #99932, #99936, #99940, … |
| maximal quotient of | #219252, #219253, #219254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15226);