Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10104 |
| 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 | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 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 — 3 generators on 32 points:
(1,2)(3,10)(4,6)(5,15)(7,9)(8,11)(12,16)(13,14)(19,25)(20,27)(22,30)(23,28)(26,29)(31,32);(1,3,6,12)(2,8,4,14)(5,13,7,11)(9,16,15,10)(17,18,21,24)(19,22,26,31)(20,28,27,23)(25,32,29,30);(1,4,6,2)(3,10,12,16)(5,15,7,9)(8,11,14,13)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,27)(28,30)Polycyclic presentation:
2^8|b^2=gh,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=eh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10111, #236433, #236440, … |
| maximal normal subgroup of | C2×#10111, #236433, #236440, … |
| maximal quotient of | #102597, #102598, #102610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3099);