Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
267 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,13)(7,14,16,19)(8,12,9,17)(11,18,20,15)(21,22,26,30)(23,31,34,35)(24,27,25,29)(28,32,36,33);(1,2)(3,4)(5,7,10,16)(6,11,13,20)(8,15,9,18)(12,19,17,14)(22,27)(24,25)(28,35)(29,30)(31,36)(32,33);(2,4)(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,23,26,34)(22,28,30,36)(24,33,25,32)(27,35,29,31)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,e^b=eh,d^c=dh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10061, #235665, #235678, … |
| maximal normal subgroup of | C2×#10061, #235665, #235678, … |
| maximal quotient of | #102941, #102943, #103048, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3049);