Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #394 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1440 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
410 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 109 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 — 3 generators on 32 points:
(1,2)(3,13,17,8)(4,18)(5,12)(6,11)(7,10)(9,20)(14,30,31,26)(15,24,16,25)(19,29,32,23)(21,28)(22,27);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,26)(10,13)(11,25)(12,24)(14,20)(18,23)(21,32)(22,31)(27,30)(28,29);(1,4)(2,9,10,27)(3,14)(5,20)(6,22)(7,21)(8,23,13,29)(11,18,12,28)(15,19)(16,32)(17,31)(24,26,25,30)Polycyclic presentation:
2^8|a^2=g,c^2=d,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,g^b=gh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11186, #359161, #359903, … |
| maximal normal subgroup of | C2×#11186, #359161, #359903, … |
| maximal quotient of | #118688, #118689, #118781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4174);