Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
456 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 456 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) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,22,25,23,26,21,30)(19,27,28,31,20,24,29,32);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19)(18,24)(20,23)(21,28)(22,29)(25,31)(26,27)(30,32);(3,9)(6,12)(10,15)(13,16)(17,20,23,19)(18,24,26,27)(21,28,22,29)(25,32,30,31);(2,7)(6,12)(8,14)(13,16)(17,21,23,22)(18,25,26,30)(19,28,20,29)(24,31,27,32)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=g,f^2=h,b^a=be,d^a=dh,e^a=eg,c^b=ch,f^b=fg,d^c=dg46 conjugacy classes — view the full character table.
| maximal subgroup of | #139445, #139447, #139465, … |
| maximal normal subgroup of | #139445, #139447, #139465, … |
| maximal quotient of | #162954, #162962, #162970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10397);