Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#538 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
539 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 32 points:
(1,2,6,11)(3,13,16,22)(4,9,5,19)(7,12,17,10)(8,20,25,14)(15,29,30,28)(18,27,21,26)(23,31,24,32);(1,3)(2,8)(4,14)(5,20)(6,16)(7,15)(9,22)(10,23)(11,25)(12,24)(13,19)(17,30)(18,31)(21,32)(26,29)(27,28);(1,4,17,21)(2,9,10,26)(3,14,30,32)(5,7,18,6)(8,22,23,29)(11,19,12,27)(13,24,28,25)(15,31,16,20);(2,10)(8,23)(9,26)(11,12)(13,28)(19,27)(22,29)(24,25)Polycyclic presentation:
2^8|a^2=g,c^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #245159, #245160, #245166, … |
| maximal normal subgroup of | #245159, #245160, #245166, … |
| maximal quotient of | #168508, #168514, #168517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11317);