Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10128 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
242 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(4,5)(8,15)(9,10)(11,16)(12,14)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,2)(3,11)(4,10)(5,7)(6,9)(8,12)(13,15)(14,16)(20,29)(23,28)(25,32)(27,30);(1,3,6,13)(2,8,9,16)(4,12,5,14)(7,11,10,15)(17,19)(18,22)(20,30)(21,26)(23,32)(24,31)(25,28)(27,29)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,e^b=egh,f^b=fg,d^c=dh,e^c=eh,f^c=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10137, #236716, #236727, … |
| maximal normal subgroup of | C2×#10137, #236716, #236727, … |
| maximal quotient of | #102654, #102673, #102675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3125);