Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
463 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 463 conjugacy classes of subgroups — too many to draw. Only its 139 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)(7,11)(8,9)(10,12)(13,14,17,21,16,20,23,27)(15,22,25,28,24,18,26,19);(6,9)(10,11)(13,15,23,26,16,24,17,25)(14,18,27,28,20,22,21,19);(2,4)(13,15,16,24)(14,19,20,28)(17,26,23,25)(18,27,22,21);(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #288774, #288780, #288810, … |
| maximal normal subgroup of | #288774, #288780, #288810, … |
| maximal quotient of | #320223, #320236, #320245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21407);