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) = C23×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 | |
717 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 717 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:
(7,11)(10,12)(13,14)(15,17)(16,18)(19,20)(21,22,24,26)(23,27,28,25);(1,2,3,4)(6,9)(10,12)(14,15)(18,19)(23,28)(25,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,18,20,19)(21,23,24,28)(22,25,26,27);(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^8|a^2=ef,b^2=egh,c^2=f,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 | #104757, #104758, #104759, … |
| maximal normal subgroup of | #104757, #104758, #104759, … |
| maximal quotient of | #155715, #155859, #155954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9812);