Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C2 | C2×D42 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1308 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 1308 conjugacy classes of subgroups — too many to draw. Only its 485 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) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,13)(4,15)(5,9)(6,10)(7,11)(8,12)(14,29)(16,18)(17,30)(19,23)(20,32)(21,26)(22,28)(24,27)(25,31);(1,3)(2,4)(5,20)(6,15)(7,23)(8,16)(9,25)(10,18)(11,13)(12,19)(14,21)(17,27)(22,30)(24,31)(26,32)(28,29);(1,4)(2,3)(5,17)(6,18)(7,13)(8,19)(9,14)(10,15)(11,23)(12,16)(20,27)(21,25)(22,32)(24,29)(26,30)(28,31);(1,5)(2,9)(3,14)(4,17)(6,21)(7,24)(8,22)(10,26)(11,27)(12,28)(13,29)(15,30)(16,31)(18,25)(19,32)(20,23);(1,6,8,7)(2,10,12,11)(3,15,16,23)(4,18,19,13)(5,21,22,24)(9,26,28,27)(14,30,31,20)(17,25,32,29)Polycyclic presentation:
2^8|e^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,d^b=df,f^b=fh,g^c=gh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152184, #499785, #499786, … |
| maximal normal subgroup of | #152184, #499785, #499786, … |
| maximal quotient of | #6358101, #6358110, #6358127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54088);