Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
315 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 107 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)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18)(19,26)(20,25)(21,23)(22,29)(24,27)(28,32)(30,31);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(18,23)(24,31)(25,29)(26,32);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(17,19)(18,24)(20,27)(21,28)(22,30)(23,31)(25,32)(26,29);(3,10)(8,15)(11,14)(12,16)(17,20,21,22)(18,25,23,29)(19,27,28,30)(24,32,31,26)Polycyclic presentation:
2^8|b^2=egh,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^c=dg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #138555, #138564, #489324, … |
| maximal normal subgroup of | #138555, #138564, #489324, … |
| maximal quotient of | #189583, #197775, #202771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12213);