Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
229 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(2,4)(5,6,9,13,11,14,10,19)(7,15,16,20,8,17,18,12)(21,22)(23,29,31,36)(24,26)(25,28)(27,30,34,33)(32,35);(1,2,3,4)(5,7,11,8)(6,12,14,20)(9,18,10,16)(13,17,19,15)(22,26)(25,32)(27,34)(30,33);(5,8,11,7)(6,12,14,20)(9,18,10,16)(13,15,19,17)(21,23)(22,27)(24,31)(25,33)(26,34)(28,36)(29,35)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94785, C2×#9004, #194477, … |
| maximal normal subgroup of | #94785, C2×#9004, #194477, … |
| maximal quotient of | #99540, #99546, #99646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1992);