Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 403 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:
(5,6,9,11)(7,8,12,10)(13,14,16,20,17,21,24,19)(15,18,22,26,23,27,28,25);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,15,17,23)(14,18,21,27)(16,22,24,28)(19,25,20,26);(2,4)(14,19)(16,24)(18,25)(20,21)(22,28)(26,27);(5,8,9,10)(6,7,11,12)(14,20)(16,24)(18,26)(19,21)(22,28)(25,27)Polycyclic presentation:
2^8|a^2=ef,b^2=eh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125745, #126086, #126351, … |
| maximal normal subgroup of | #125745, #126086, #126351, … |
| maximal quotient of | #345598, #345614, #345627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24233);