Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
433 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 433 conjugacy classes of subgroups — too many to draw. Only its 143 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)(5,6,8,10)(7,9,11,12)(13,14,17,20)(15,21,24,28)(16,25,22,19)(18,26,27,23);(5,7)(6,9)(8,11)(10,12)(13,15,22,26,17,24,16,23)(14,18,25,28,20,27,19,21);(2,4)(7,11)(9,12);(13,15,22,26,17,24,16,23)(14,18,25,28,20,27,19,21)Polycyclic presentation:
2^8|a^2=gh,b^2=eh,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #122699, #122715, #122740, … |
| maximal normal subgroup of | #122699, #122715, #122740, … |
| maximal quotient of | #314657, #314662, #314668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20650);