Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10212 |
| 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 | |
234 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,15,20)(11,21,24,35)(12,25,28,33)(13,19,14,18)(16,29,31,23)(17,26,34,27)(22,30,38,32)(36,39,40,37);(2,5)(3,8)(6,7)(9,11)(10,16)(12,22)(13,29)(14,23)(15,24)(17,32)(18,35)(19,21)(20,31)(25,36)(26,39)(27,37)(28,38)(30,34)(33,40);(1,2,4,7)(3,6,8,5)(9,12,15,28)(10,17,20,34)(11,22,24,38)(13,27,14,26)(16,30,31,32)(18,25,19,33)(21,36,35,40)(23,39,29,37)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,f^b=fgh,d^c=dh,e^c=eh,f^c=fgh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94659, #94661, #151169, … |
| maximal normal subgroup of | #94659, #94661, #151169, … |
| maximal quotient of | #102510, #102514, #102517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3204);