Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
388 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 123 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)(7,9)(8,10)(11,12)(13,14,18,22)(15,20,24,28)(16,26,25,19)(17,27,23,21);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,23,16,18,24,17,25)(14,19,27,28,22,26,21,20);(2,4)(6,10)(9,12)(15,24)(16,25)(19,26)(20,28);(13,16,23,24,18,25,17,15)(14,20,27,19,22,28,21,26)Polycyclic presentation:
2^8|a^2=h,b^2=efh,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #501871, #504386, #504523, … |
| maximal normal subgroup of | #501871, #504386, #504523, … |
| maximal quotient of | #281328, #281334, #281340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18037);