Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
649 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 649 conjugacy classes of subgroups — too many to draw. Only its 175 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:
(2,4)(7,11)(9,12)(14,17)(15,19)(23,27)(25,28);(13,14)(15,18)(16,17)(19,20)(21,22,24,26)(23,25,27,28);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14)(15,20)(16,17)(18,19)(21,22)(23,25)(24,26)(27,28);(5,7)(6,9)(8,11)(10,12)(13,15,16,19)(14,18,17,20)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^8|b^2=g,c^2=egh,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #255256, #255276, #255306, … |
| maximal normal subgroup of | #255256, #255276, #255306, … |
| maximal quotient of | #155823, #155831, #155865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10054);