Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | Aut(G) = C2×#72296209 (order 8192) · Inn(G) = #120 · Out(G) = C2×#2846 |
| 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 | |
242 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(5,6,11,16)(7,17,8,19)(9,15,10,14)(12,20,13,18)(21,22,26,24)(23,29,32,36)(25,33,35,28)(27,34,30,31);(1,2)(3,4)(5,7)(6,12)(8,11)(9,20)(10,18)(13,16)(14,19)(15,17)(21,23,26,32)(22,27,24,30)(25,34,35,31)(28,36,33,29);(2,4)(5,8,11,7)(6,13,16,12)(9,20,10,18)(14,19,15,17)(21,24,26,22)(23,30,32,27)(25,28,35,33)(29,34,36,31)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=cf,d^b=dg,e^b=eh,d^c=dh,e^c=egh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10084, #233637, #233639, … |
| maximal normal subgroup of | C2×#10084, #233637, #233639, … |
| maximal quotient of | #102511, #102522, #102530, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3072);