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 | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
263 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 263 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,14)(7,15,17,20)(8,13,9,12)(11,18,19,16)(21,22,25,28)(23,29,31,36)(24,32,34,27)(26,33,35,30);(1,2)(3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19)(21,23,25,31)(22,26,28,35)(24,33,34,30)(27,36,32,29);(2,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,15)(13,20)(14,19)(24,34)(27,32)(29,36)(30,33)Polycyclic presentation:
2^8|a^2=g,b^2=h,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×#10083, #233634, #233637, … |
| maximal normal subgroup of | C2×#10083, #233634, #233637, … |
| maximal quotient of | #102509, #102520, #102526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3071);