Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1055 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1055 conjugacy classes of subgroups — too many to draw. Only its 567 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 — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(1,2,3,4)(15,21)(18,24)(22,27)(25,28);(14,19)(18,24)(20,26)(25,28);(13,16,17,23)(14,20,19,26)(15,22,21,27)(18,25,24,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,e^2=h,b^a=bf,c^a=cg,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #111255, #111273, #112628, … |
| maximal normal subgroup of | #111255, #111273, #112628, … |
| maximal quotient of | #515058, #515070, #515162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27630);