Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×D8 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
303 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 303 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,9,10,16)(4,13,14,7)(6,11,15,12)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,3)(2,6)(4,11)(5,10)(7,16)(8,15)(9,13)(12,14);(3,10)(6,15)(9,16)(11,12)(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(3,11)(4,14)(6,16)(7,13)(9,15)(10,12)Polycyclic presentation:
2^8|a^2=gh,c^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=de,e^b=eg,f^c=fh,e^d=eg43 conjugacy classes — view the full character table.
| maximal subgroup of | #123335, #123343, #123351, … |
| maximal normal subgroup of | #123335, #123343, #123351, … |
| maximal quotient of | #220780, #220949, #225532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12915);