Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#459 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
422 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,20)(17,19)(21,25)(22,27)(23,26)(24,28);(6,9)(10,12)(13,15)(14,18)(16,21)(17,23)(19,26)(20,25)(22,24)(27,28);(2,4)(7,11)(10,12)(15,21)(16,24)(18,25)(20,28)(22,23)(26,27);(5,7,8,11)(6,10,9,12)(14,19)(18,26)(20,28)(25,27)Polycyclic presentation:
2^8|a^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #273283, #273285, #273300, … |
| maximal normal subgroup of | #273283, #273285, #273300, … |
| maximal quotient of | #330559, #330560, #330561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22822);