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 | |
653 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 653 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(3,7)(9,10,13,11)(12,14,16,15)(17,18,20,22)(19,21,23,24)(26,29)(27,31);(2,5)(6,8)(10,11)(14,15)(19,23)(21,24)(25,26,28,29)(27,30,31,32);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,14,16,15)(17,19)(18,21)(20,23)(22,24)(25,26)(27,32)(28,29)(30,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^8|a^2=fg,b^2=egh,c^2=ef,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 | #104769, #104770, #104771, … |
| maximal normal subgroup of | #104769, #104770, #104771, … |
| maximal quotient of | #155715, #155872, #155894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9891);