Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
315 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 163 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,16)(17,18,23,19,22,25,21,27)(20,24,28,26,29,32,30,31);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,23,25,22,27,21,18)(20,26,28,32,29,31,30,24);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(17,20,22,29)(18,24,25,32)(19,26,27,31)(21,30,23,28);(9,12,13,16)(10,14,11,15)(17,21,22,23)(18,19,25,27)(20,28,29,30)(24,31,32,26)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ceh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22316, #6376691, #6376693, … |
| maximal normal subgroup of | C2×#22316, #6376691, #6376693, … |
| maximal quotient of | #355271, #355403, #355861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15304);