Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| 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 | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
438 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 438 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:
(2,4)(5,6)(7,11)(8,10)(9,12)(14,19)(20,27)(21,22)(23,28);(5,7,8,12)(6,9,10,11)(13,14,16,21,17,19,18,22)(15,23,26,20,25,28,24,27);(1,2,3,4)(13,15,17,25)(14,20,19,27)(16,24,18,26)(21,23,22,28);(6,10)(9,11)(13,16,17,18)(14,21,19,22)(15,24,25,26)(20,23,27,28)Polycyclic presentation:
2^8|b^2=eg,c^2=fh,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #488705, #488706, C2×#28564, … |
| maximal normal subgroup of | #488705, #488706, C2×#28564, … |
| maximal quotient of | #324130, #324132, #324136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21552);