Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1211 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1211 conjugacy classes of subgroups — too many to draw. Only its 487 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) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(9,12)(16,23)(20,25)(22,27)(26,28);(1,2,3,4)(5,6,8,10)(7,9,11,12)(14,18)(15,21)(20,25)(22,27);(5,7,8,11)(6,9,10,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28);(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)Polycyclic presentation:
2^8|b^2=fg,c^2=gh,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #112563, #112595, #112611, … |
| maximal normal subgroup of | #112563, #112595, #112611, … |
| maximal quotient of | #516427, #516460, #516584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28326);