Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| 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 | C42 | C24 |
| prime radical | #3559 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
440 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 440 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(20,29)(23,28)(25,32)(27,30);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,19)(18,22)(20,30)(21,26)(23,32)(24,31)(25,28)(27,29);(2,8)(6,12)(7,16)(14,15)Polycyclic presentation:
2^8|a^2=gh,c^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | #250840, #250842, #250851, … |
| maximal normal subgroup of | #250840, #250842, #250851, … |
| maximal quotient of | #166300, #166347, #166428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12011);